Texture segmentation plays a crucial role in the domain of image analysis and its recognition. Noise is inextricably linked to images, just like it is with every signal received by sensing, which has an impact on how well the segmentation process performs in general. Recent literature reveals that the research community has started recognizing the domain of noisy texture segmentation for its work towards solutions for the automated quality inspection of objects, decision support for biomedical images, facial expressions identification, retrieving image data from a huge dataset and many others. Motivated by the latest work on noisy textures, during our work being presented here, Brodatz and Prague texture images are contaminated with Gaussian and salt-n-pepper noise. A three-phase approach is developed for the segmentation of textures contaminated by noise. In the first phase, these contaminated images are restored using techniques with excellent performance as per the recent literature. In the remaining two phases, segmentation of the restored textures is carried out by a novel technique developed using Markov Random Fields (MRF) and objective customization of the Median Filter based on segmentation performance metrics. When the proposed approach is evaluated on Brodatz textures, an improvement of up to 16% segmentation accuracy for salt-n-pepper noise with 70% noise density and 15.1% accuracy for Gaussian noise (with a variance of 50) has been made in comparison with the benchmark approaches. On Prague textures, accuracy is improved by 4.08% for Gaussian noise (with variance 10) and by 2.47% for salt-n-pepper noise with 20% noise density. The approach in the present study can be applied to a diversified class of image analysis applications spanning a wide spectrum such as satellite images, medical images, industrial inspection, geo-informatics, etc.
Citation: Sanjaykumar Kinge, B. Sheela Rani, Mukul Sutaone. Restored texture segmentation using Markov random fields[J]. Mathematical Biosciences and Engineering, 2023, 20(6): 10063-10089. doi: 10.3934/mbe.2023442
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Texture segmentation plays a crucial role in the domain of image analysis and its recognition. Noise is inextricably linked to images, just like it is with every signal received by sensing, which has an impact on how well the segmentation process performs in general. Recent literature reveals that the research community has started recognizing the domain of noisy texture segmentation for its work towards solutions for the automated quality inspection of objects, decision support for biomedical images, facial expressions identification, retrieving image data from a huge dataset and many others. Motivated by the latest work on noisy textures, during our work being presented here, Brodatz and Prague texture images are contaminated with Gaussian and salt-n-pepper noise. A three-phase approach is developed for the segmentation of textures contaminated by noise. In the first phase, these contaminated images are restored using techniques with excellent performance as per the recent literature. In the remaining two phases, segmentation of the restored textures is carried out by a novel technique developed using Markov Random Fields (MRF) and objective customization of the Median Filter based on segmentation performance metrics. When the proposed approach is evaluated on Brodatz textures, an improvement of up to 16% segmentation accuracy for salt-n-pepper noise with 70% noise density and 15.1% accuracy for Gaussian noise (with a variance of 50) has been made in comparison with the benchmark approaches. On Prague textures, accuracy is improved by 4.08% for Gaussian noise (with variance 10) and by 2.47% for salt-n-pepper noise with 20% noise density. The approach in the present study can be applied to a diversified class of image analysis applications spanning a wide spectrum such as satellite images, medical images, industrial inspection, geo-informatics, etc.
For each 1≤k≤N, the k-Hessian operator acting on u∈C2(Ω) with Ω⊆RN open is defined by
Sk(D2u):=σk(λ(D2u)):=∑1≤i1<⋯<ik≤Nλi1(D2u)⋯λik(D2u) | (1.1) |
where λ(D2u) indicates the N-vector of ordered eigenvalues of the Hessian matrix D2u and σk(λ) is the elementary symmetric polynomial, which is homogeneous of degree k. For example, one has
S1(D2u)=Δu=tr(D2u) andSN(D2u)=det(D2u). |
Before describing the scope of this paper, let us mention that, for each k>1, the k-Hessian is fully nonlinear and is (degenerate) elliptic only when constrained in a suitable sense. More precisely, in general one does not have
X≤Y⇒Sk(X)≤Sk(Y) | (1.2) |
if X,Y are free to range over all of S(N), the space of N×N symmetric matrices. However, one will have (1.2) if one constrains X (and hence Y) to belong to
Σk:={A∈S(N): λ(A)∈¯Γk} | (1.3) |
where
Γk:={λ∈RN: σj(λ)>0, j=1,…,k}. | (1.4) |
This leads us to work in the context of admissible viscosity solutions, using the notion of elliptic sets as introduced by Krylov [23]. This notion has given rise to the development of an organic theory of viscosity solutions with admissibility constraints beginning with Harvey-Lawson [11] for pure second order operators on Euclidean space and generalized to general operators on Riemannian manifolds in [13]. In the terminology of [7], Σk is a (constant coefficient) pure second order subequation constraint set which requires that Σk⊂S(N) be a proper closed subset satisfying the positivity property
A∈Σk ⇒ A+P∈Σk for each P∈P, P:={P∈S(N): P≥0} | (1.5) |
and the topological property
Σk=¯Σ∘k, | (1.6) |
which is the closure in S(N) of the interior of Σk.
In the nonlinear potential theory language of [11], one would say that u∈USC(Ω) is Σk-subharmonic at x0 if u is k-convex at x0∈Ω in the sense of Definition 2.1. Moreover, one has the following coherence property: if u∈USC(Ω) is twice differentiable in x0 then
u is Σk-subharmonic at x0 ⇔ D2u(x0)∈Σk; | (1.7) |
that is, the classical and viscosity notions of k-convexity coincide at points of twice differentiability. We notice that the reverse implication (⇐) depends on the positivity property (1.5) of Σk. The topological property (1.6) is sufficient for the local construction of classical strict subsolutions φ; that is, Sk(D2φ)<0. It also plays and important role in the Harvey-Lawson duality which leads to an elegant formulation of supersolutions by duality (see Remark 2.12 and Proposition 3.2 of [7] for details).
The main purpose of this paper is to study the principal eigenvalue associated to the operator −Sk with homogeneous Dirichlet data in bounded domains Ω by using maximum principle methods. We shall consider Ω⊂RN whose boundaries are of class C2 and (k−1)-convex. The notion of principal eigenvalue that we will define is inspired by that of Berestycki, Nirenberg and Varadhan in their groundbreaking paper [1].
More precisely, for each k∈{1,…,N} fixed, using the notion of k-convexity in a viscosity sense, we introduce
Φ−k(Ω):={ψ∈USC(Ω):ψ is k-convex and negative in Ω}. |
We can then consider
Λk:={λ∈R: ∃ψ∈Φ−k(Ω) with Sk(D2ψ)+λψ|ψ|k−1≥0 in Ω}, |
where the inequality above is again in the admissible viscosity sense, and define our candidate for a (generalized) principal eigenvalue by
λ−1:=supΛk. |
We will prove that λ−1 is an upper bound for the validity of the minimum principle in Ω; that is, we will show that for any λ<λ−1 the operator F[⋅]=Sk(D2⋅)+λ⋅|⋅|k−1 satisfies the minimum principle in Ω (see Theorem 5.4). Moreover, in Theorem 6.6 we will show that λ−1 is actually an eigenvalue for the operator −Sk in the sense that there exists ψ1<0 in Ω such that
{Sk(D2ψ1)+λ−1ψ1|ψ1|k−1=0in Ωψ1=0on ∂Ω. |
Notice that is the linear case k=1 this conforms to the usual notion of eigenvalue. If k is odd integer, then Sk is an odd operator and we would have a maximum principle characterization for λ+1; that is, a principal eigenvalue which corresponds to a positive concave principal eigenfunction.
We should mention that the k-Hessians are variational and hence it is possible to give a variational characterization of the principal eigenvalue through a generalized Rayleigh quotient. This was done by Lions [26] for k=N and by Wang [32] for general k. Using the minimum principle, we prove that both characterizations coincide (see Corollary 5.7). Hence the existence of the eigenfunction corresponding to the so-called eigenvalue is just a consequence of this equality, using the results in [26,32].
Nonetheless, we wish to give a proof of the existence that is independent of the variational characterization. Indeed, the interest in defining the principal eigenvalue by way of (5.4) is twofold. On the one hand, it allows one to prove the minimum principle for λ below λ−1 while on the other hand, it strongly suggests that it may be possible to extend the results to a class of fully non linear operators that are not variational, but which may include the k-Hessians. For example, with 0<α≤β and Sα,β={A∈S(N): αI≤A≤βI}, one might consider the operators defined by
σ+k(D2u):=supA∈Sα,βσk(λ(AD2u)). |
This will be the subject of a subsequent paper.
We should also mention that the challenges for a viscosity solution approach to the existence of a solution ψ1 to the eigenvalue problem
{Sk(D2ψ1)+λ−1ψ1|ψ1|k−1=0in Ωψ1=0on ∂Ω | (1.8) |
include the lack of global monotonicity in the Hessian D2ψ1 and the "wrong" monotonicity in ψ1. The argument will involve an iterative construction that has its origins in [2,3] and was used with success in degenerate settings in [4]. At some point in the argument, a compactness property is needed for the sequence of k-convex solutions to the approximating equations. The needed compactness property follows from a global Hölder estimate on the sequence of approximating solutions. Unfortunately, using merely maximum principle methods, we are able to prove the estimate only in the case when k>N2. See Remark 6.4 for a discussion of this point, including the use of some measure theoretic techniques (to augment the maximum principle techniques developed herein) in order establish existence in the nonlinear range 1<k≤N2.
In all that follows, Ω will be a bounded open domain in RN, S(N) is the space of N×N symmetric matrices with real entries and
λ1(A)≤⋯≤λN(A) | (2.1) |
are the N eigenvalues of A∈S(N) written in increasing order. The spaces of upper, lower semi-continuous functions on Ω taking values in [−∞,+∞),(−∞,+∞] will be denoted by USC(Ω),LSC(Ω) respectively.
We begin by describing the subset Σk⊂S(N) which serves to define k-convex functions by way of a viscosity constraint on the Hessian of u∈USC(Ω). For k∈{1,…,N} denote by
σk(λ):=∑1≤i1<⋯<ik≤Nλi1⋯λik | (2.2) |
the kth-elementary symmetric function of λ∈RN and consider the open set defined by
Γk:={λ∈RN: σj(λ)>0, j=1,…,k}, | (2.3) |
which is clearly a cone with vertex at the origin and satisfies
Γ+⊂ΓN⊂ΓN−1⋯⊂Γ1 | (2.4) |
where Γ+:={λ∈RN: λi>0 for each i} is the positive cone in RN. The obvious inclusion Γ+⊂ΓN is, in fact, an identity (2.14), as will be discussed below.
Additional fundamental properties of the cones Γk are most easily seen by using other alternate characterizations of (2.3). First, the homogeneous polynomials (2.2) are examples of hyperbolic polynomials in the sense of Gårding. More precisely, for each fixed k, σk is homogeneous of degree k and is hyperbolic in the direction a=(1,…,1)∈RN; that is, the degree k polynomial in t∈R defined by
σk(ta+λ)=∑1≤i1<⋯<ik≤N(t+λi1)⋯(t+λik) | (2.5) |
has k real roots for each λ∈RN. The cone Γk can be defined as
Γk is the connected component of {λ∈RN:σk(λ)>0} containing (1,…,1), | (2.6) |
from which it follows that
λ∈∂Γk ⇒ σk(λ)=0. | (2.7) |
The form (2.6) of Γk corresponds to Gårding's original definition for a general hyperbolic polynomial for which the form (2.3) results in the special case of the kth- elementary symmetric function σk. The reader might wish to consult section 1 of Caffarelli-Nirenberg-Spruck [5].
Gårding's beautiful theory of hyperbolic polynomials in [9] applied to σk includes two important consequences; namely convexity
the cone Γk is convex | (2.8) |
and (strict) monotonicity
σk(λ+μ)>σk(λ) for each λ,μ∈Γk. | (2.9) |
Since Γ+⊂Γk for each k and Γk is a cone, one has that Γk is a monotonicity cone for itself; that is,
Γk+Γk⊂Γk | (2.10) |
and, in particular,
Γ++Γk⊂Γk, | (2.11) |
which, with the monotonicity (2.9) for μ∈Γ+, gives rise to the degenerate ellipticity of k-Hessian operators as it will be recalled below.
Moreover, Korevaar [21] has characterized Γk as
Γk={λ∈RN: σk(λ)>0,∂σk∂λi1(λ)>0,…,∂k−1σk∂λi1⋯∂λik−1(λ)>0 for all {1≤i1<⋯<ik−1≤N}} | (2.12) |
which implies that
N−k+1∑j=1λij>0 on Γk for all {1≤i1<⋯<ik−1≤N}. | (2.13) |
When k=N this says that each λi>0 for each i and hence
ΓN=Γ+. | (2.14) |
Additional characterizations of Γk, interesting and useful identities and inequalities involving σk can be found in [5,18,19,20,25,30,31,33]. For a modern and self-contained account of Gårding's theory and its relation to the Dirichlet problem one can consult [12] and [14].
Clearly the closed Gårding cone ¯Γk is also convex with
¯Γ+=¯ΓN⊂¯ΓN−1⋯⊂¯Γ1 | (2.15) |
and by continuity the monotonicity properties extend to say
¯ΓN+¯Γk⊂¯Γk and σk(λ+μ)≥σk(λ) for each λ∈¯Γk,μ∈¯ΓN. | (2.16) |
For A∈S(N) denote by λ(A):=(λ1(A),…,λN(A)) the vector of eigenvalues (2.1) and define the k-convexity constraint set by
Σk:={A∈S(N): λ(A)∈¯Γk}. | (2.17) |
For u∈C2(Ω), one says that u is k-convex on Ω if
D2u(x)∈Σk for each x∈Ω | (2.18) |
and we will say that u∈C2(Ω) is strictly k-convex on Ω if D2u(x) lies in the interior Σ∘k of Σk for each x∈Ω. Notice that
ΣN⊂ΣN−1⋯⊂Σ1 | (2.19) |
where
Σ1=H:={A∈S(N): tr(A)≥0} and ΣN=P:={A∈S(N): λ1(A)≥0}; | (2.20) |
that is, 1-convex functions are classically subharmonic (with respect to the Laplacian) and N-convex functions are ordinary convex functions. This consideration carries over to u∈USC(Ω) where one defines k-convexity by interpreting (2.18) in the viscosity sense.
Definition 2.1. Given u∈USC(Ω), we say that u is k-convex at x0∈Ω if for every φ which is C2 near x0
u−φ has a local maximum in x0 ⇒ D2φ(x0)∈Σk. | (2.21) |
We say that u is k-convex on Ω if this pointwise condition holds for each x0∈Ω.
Remark 2.2. In the viscosity language of superjets, the condition (2.21) means
(p,A)∈J2,+u(x0) ⇒ A∈Σk, | (2.22) |
and there are obviously many equivalent formulations. For example, one can restrict to upper test functions φ which are quadratic and satisfy φ(x0)=u(x0). See Appendix A of [7] for a discussion of this point in the context of viscosity subsolutions with admissibility constraints. Moreover, since Σk is closed, one also has
(p,A)∈¯J2,+u(x0) ⇒ A∈Σk; | (2.23) |
where, as usual, (p,A)∈¯J2,+u(x0) means that there exists {xk,pk,Ak}k∈N such that (pk,Ak)∈J2,+u(xk) and (xk,pk,Ak)→(x0,p,A) as k→+∞.
The k-convex constraint Σk will be used as an admissibility constraint for the solutions of k-Hessian equations considered here. One defines the k-Hessian operator by
Sk(A):=σk(λ(A))for A∈S(N) and k∈{1,…,N}, | (2.24) |
where λ(A) is the vector of eigenvalues of A and σk is given by (2.2). Notice that
Sj(A)≥0 for each A∈Σk and each j=1,…k. | (2.25) |
In particular Sk is non-negative on Σk. Important special cases are
S1(A):=tr(A)andSN(A):=det(A). | (2.26) |
The following Lemma gives the fundamental structural properties of Σk and Sk.
Lemma 2.3. For each fixed k∈{1,…,N}, one has the following properties.
(a)Σk is a closed convex cone with vertex at the origin.
(b)Σk is an elliptic set; that is, Σk⊊S(N), is closed, non empty and satisfies the positivity property
A∈Σk ⇒ A+P∈Σk foreach P∈P, | (2.27) |
where P are the non negative matrices as defined in (2.20).
(c)Σk satisfies the topological property
Σk=¯Σ∘k, | (2.28) |
where Σ∘k:={A∈S(N):λ(A)∈Γk} is the interior of Σk.
(d) The k-Hessian is increasingalongΣk_; that is,
Sk(A+P)≥Sk(A) foreach A∈Σk and P∈P. | (2.29) |
Moreover, the inequality in (2.29) is strict if P∈P∘; that is, if P>0 in S(N).
Proof. Part (a) follows from the corresponding properties for ¯Γk. For the claims in part (b), Σk is closed by part (a). Each Σk is non empty since Σk⊃ΣN=P as noted in (2.19) and (2.20). Clearly Σ1⊊S(N) and hence the same is true for the other values of k by (2.19). The property (2.28) also clearly holds since Σk is a closed convex cone with non-empty interior.
For the property (2.27), if A∈Σk and P≥0 then λi(A+P)≥λi(A) for each i=1,…,N and hence using (2.16) with λ:=λ(A)∈¯Γk and μ:=λ(A+P)−λ(A)∈¯ΓN gives for each j=1,…,k
σj(λ(A+P))=σj(λ(A)+λ(A+P)−λ(A))≥σj(λ(A))≥0, | (2.30) |
which gives the positivity property (2.27).
The monotonicity formula (2.29) follows from (2.30) with j=k and the definition of the k-Hessian operator Sk(A):=σk(λ(A)). When P∈P∘, λi(A+P)>λi(A) for each i=1,…,N and the inequality in (2.29) becomes strict.
As noted in the introduction, the notion of elliptic sets was introduced by Krylov [23] and starting with the groundbreaking paper Harvey-Lawson [11], has given birth to a nonlinear potential theory approach to viscosity solutions with admissibility constraints. In the terminology of [7], Σk is a (constant coefficient) pure second order subequation constraint set which requires that Σk⊂S(N) be a proper closed subset satisfying the positivity property (2.27) and the topological property (2.28). Moreover, being also a convex cone, Σk is a monotonicity cone subequation and Σk is the maximal monotonicity cone for both Σk and for its dual monotonicity constraint set ˜Σk as defined below in (2.36) (see Proposition 4.5 of [7]). One has reflexivity (≈Σk=Σk) if topological property (2.28) holds (see Proposition 3.2 of [7], for example). The following fact, mentioned in the introduction, is worth repeating here.
Remark 2.4. In the nonlinear potential theory language of [11], one says that u∈USC(Ω) is Σk-subharmonic at x0 if u is k-convex at x0∈Ω in the sense of Definition 2.1. In addition, one has the coherence property: if u∈USC(Ω) is twice differentiable in x0 then
u is Σk-subharmonic at x0 ⇔ D2u(x0)∈Σk. | (2.31) |
We now turn to the definition of Σk-admissible viscosity subsolutions and supersolutions for the type of equations involving k-Hessian operators Sk that we will treat. The definitions make sense for any constant coefficient pure second order subequation. The main point is to indicate the role of the k-convexity constraint Σk which insures the positivity property for Sk, which corresponds to the degenerate ellipticity of Sk when the Hessian is constrained to Σk.
Definition 2.5. Let Ω⊂RN be an open set and let f:Ω×R×RN→R be continuous. Consider the equation
Sk(D2u)−f(x,u,Du)=0. | (2.32) |
(a) A function u∈USC(Ω) is said to be a Σk-admissible subsolution of (2.32) at x0∈Ω if for every φ which is C2 near x0
u−φ has a local maximum in x0 ⇒ Sk(D2φ(x0))−f(x0,u(x0),Dφ(x0))≥0and D2φ(x0)∈Σk. | (2.33) |
(b) A function u∈LSC(Ω) is said to be a Σk-admissible supersolution of (2.32) at x0∈Ω if for every φ which is C2 near x0
u−φ has a local minimum in x0 ⇒ Sk(D2φ(x0))−f(x0,u(x0),Dφ(x0))≤0or D2φ(x0)∉Σk. | (2.34) |
(c) A function u∈C(Ω) is said to be a Σk-admissible solution of (2.32) at x0 if both (2.33) and (2.34) hold.
One says that u is a Σk admissible (sub-, super-) solution on Ω if the corresponding statement holds for each x0∈Ω.
A fundamental example involves f≡0.
Example 2.6. [k-convex and co-k-convex functions] By (2.33), a function u∈USC(Ω) is a Σk-admissible subsolution of
Sk(D2u)=0 in Ω | (2.35) |
precisely when u is k-convex in Ω (which is equivalent to u being Σk-subharmonic in Ω). On the other hand, Σk-admissible supersolutions of (2.35) can be stated in terms of the Dirichet dual of Harvey-Lawson [11]
˜Σk:=−(Σ∘k)c=(−Σ∘k)c, | (2.36) |
where ˜Σk is also a constant coefficient pure second order subequation. Using (2.34) and (2.36), one can show that u∈LSC(Ω) is a Σk-admissible supersolution of (2.35) if and only if
−u∈USC(Ω) is ˜Σk-subharmonic in Ω. | (2.37) |
One says that u is Σk-superharmonic in Ω and that v:=−u is a co-k-convex function in Ω. This claim follows from the Correspondence Principle in Theorem 10.14 of [7] which in our pure second order situation requires three hypotheses. The first hypothesis is that (Sk,Σk) is a compatible operator-subequation pair since Sk∈C(Σk) with
infΣkSk=0 and ∂Σk={A∈Σk:Sk(A)=0}, | (2.38) |
which follow from the definitions of Sk and Σk. The second hypothesis is that the pair is M-monotone for some convex cone subequation M, which is true for M=P in this case. Third hypothesis is that Sk is tolpologically tame which means that {A∈Σk:Sk(A)=0} has non-empty interior, which follows from the strict monotonicity of Sk in the interior of Σk.
A few additional remarks about Definition 2.5 are in order. First we note that, of course, there are various equivalent formulations in terms of different spaces of (upper, lower) test functions φ for u in x0 in the spirit of Remark 2.2.
Remark 2.7. Concerning the Σk-admissibility, notice that:
(a) the part D2φ(x0)∈Σk of the subsolution condition (2.33) is precisely (2.21) so that u is automatically k-convex in x0;
(b) the supersolution condition (2.34) can be rephrased as
u−φ has a local minimum in x0 ⇒ Sk(D2φ(x0))−f(x0,u(x0),Dφ(x0))≤0if D2φ(x0)∈Σk; | (2.39) |
that is, it is enough to use lower test functions which are k-convex in x0.
The admissible supersolution definition takes its inspiration from Krylov [23] and was developed in [6] for equations of the form F(x,D2u)=0. In the convex Monge-Ampère case k=N of (2.32), an analogous definition was given by Ishii-Lions [17]. One good way to understand the supersolution definition (2.34) (or (2.39)) was pointed out in the convex case in [17] and concerns the following coherence property.
Remark 2.8. Suppose that u∈C2(Ω) is a classical supersolution in Ω; that is,
Sk(D2u(x))−f(x,u(x),Du(x))≤0, x∈Ω. | (2.40) |
If φ∈C2(Ω) is a lower test function for u in x0 (u−φ has local minimum in x0), while one has D2u(x0)≥D2φ(x0) from elementary calculus, one cannot use this to deduce
Sk(D2φ(x0))−f(x,u(x0),Du(x0))≤0 |
unless_D2φ(x0)∈Σk.
As a final remark, we note that our main focus will be for the equation
Sk(D2u)+λu|u|k−1=0. | (2.41) |
where k∈{1,…,N} and λ∈R is a spectral parameter, which will be positive in the interesting cases and associated to (2.41) we will often have a homogeneous Dirichlet condition on ∂Ω. We will have cause to consider negative and k-convex subsolutions to (2.41) as well as non negative supersolutions. Obviously, this means using Definition 2.5 with f(x,u,Du)=−λu|u|k−1 where the positivity of λ and negativity of u is compatible with the Σk convexity of (sub)solutions u.
In order to construct suitable barriers for k-Hessian operators, we will exploit a suitable notion of strict boundary convexity which is stated in terms of the positivity of the relevant elementary symmetric function of the principal curvatures. More precisely, given Ω⊂RN a bounded domain with ∂Ω∈C2, we denote by
(κ1(y),…,κN−1(y)) with y∈∂Ω | (2.42) |
the principal curvatures (relative to the inner unit normal ν(y)) which are defined as the eigenvalues of the self-adjoint shape operator S on the tangent space T(y) defined by
S(X):=−DXν, X∈T(y). | (2.43) |
If the boundary is represented locally near a fixed point y0∈∂Ω as the graph of a suitable function ϕ, the principle curvatures κi(y0) are the eigenvalues of the Hessian of ϕ at the relevant point. This will be recalled in the next subsection (as will special coordinate systems well adapted for calculations near the boundary).
The needed concept of convexity is the following notion.*
*This is known uniform (k−1)-convexity as in the works of Trudinger beginning with [27] (see also [28]).
Definition 2.9. Let k∈{2,…,N}. Ω⊂RN with ∂Ω∈C2 is said to be strictly (k−1)-convex if†
†Here and below, we will use the same symbol σj for the jth-elementary symmetric function on RN−1 and RN.
σj(κ1(y),…,κN−1(y))>0 for each y∈∂Ω and each j=1,…k−1; | (2.44) |
that is, for each j=1,…,k−1, each jth-mean curvature is everywhere strictly positive on ∂Ω.
Notice that strict (N−1)-convexity is ordinary strict convexity of ∂Ω. One importance of this convexity is that it ensures the existence of functions which are C2, vanish on the boundary and strictly k-convex near the boundary. This fact will be used in Proposition 4.2 below and depends in part on the following fact.
Lemma 2.10. If Ω⊂RN is a bounded strictly (k−1)-convex domain with ∂Ω∈C2, then there exists R>0 such that
σj(κ1(y),…,κN−1(y),R)>0 foreach y∈∂Ω andeach j=1,…k; | (2.45) |
that is,
(κ1(y),…,κN−1(y),R)∈Γk foreach y∈∂Ω. | (2.46) |
Proof. With the conventions that σ0(⋅)=1 and σj(λ)=0 if λ∈Rn with j>n, one has the elementary identity
σj(κ1,…,κN−1,R)=Rσj−1(κ1,…,κN−1)+σj(κ1,…,κN−1), j=1,…,k. | (2.47) |
If 1≤j<k≤N, for each R>0 both terms on the right hand side of (2.47) are positive on ∂Ω by the convexity assumption (2.44). If 1≤j=k<N, the first term on the right hand side of (2.47) is positive by (2.44) and both terms are continuous functions on ∂Ω which is compact, which gives the claim (2.45) if R is large enough. In the remaining case j=k=N, the second term in the right hand side of (2.47) vanishes, while the first term is positive for every R>0 by (2.44).
We note that if ∂Ω is connected then the conclusion (2.45) holds under the weaker convexity assumption
σk(κ1,…,κN−1)>0 on ∂Ω. | (2.48) |
See Remark 1.2 of [5] for a proof of this fact, which also makes use of (2.47).
As a final consideration, we make a comparison with the natural notion of strict →Σk-convexity, as defined in section 5 of Harvey-Lawson [11]. This notion is defined in terms of an elliptic cone →Σ which is an elliptic subset of S(N) (as defined in Lemma (2.3) (a)) which is also a pointed cone in the sense that
A∈→Σ ⇔ tA∈→Σ for each t≥0. |
Given an elliptic set Σ there is an associated elliptic cone →Σ which can be defined as the closure of the set
{A∈S(N): ∃t0>0 with tA∈Σ for each t≥t0} |
It is easy to see that if Σ is an elliptic cone, then →Σ=Σ.
One says that ∂Ω is strictly →Σ-convex at x∈∂Ω if there exists a local defining function ρ for the boundary near x such that ‡
‡More precisely, ρ∈C2(Br(x)) for some r>0 and Ω∩Br(x)={y∈Br(x): ρ(y)<0} and Dρ≠0 on Br(x).
D2ρ(x)|Tx∂Ω=B|Tx∂Ω for some B∈→Σ∘, | (2.49) |
which is to say that ρ is strictly →Σ convex near x∈∂Ω. In [11], it is shown that solvability of the Dirichlet problem on Ω for Σ-harmonic functions holds if ∂Ω is strictly →Σ and →˜Σ convex where ˜Σ=−(Σ∘)c is the Dirichlet dual of Σ (as defined in (2.36)).
Proposition 2.11. For Ω⊂RN bounded with ∂Ω∈C2, one has
∂Ω isstrictly (k−1)−convex ⟺ ∂Ω isstrictly →Σk and →˜Σ convex. | (2.50) |
Proof. Since Σk and ˜Σk are elliptic sets and pointed cones, they are themselves elliptic cones and hence
Σk=→Σkand˜Σk=→˜Σk. | (2.51) |
From (2.19) and (2.20) one has for each k∈{1,…,N}
P=ΣN⊂Σk⊂Σ1=H | (2.52) |
and by the definition of the dual one also has
H=˜H⊂˜Σ1⊂˜Σk⊂˜ΣN=˜P | (2.53) |
and hence
Σk⊂˜Σk for each k∈{1,…,N}. | (2.54) |
From (2.51) and (2.54) one has that →Σk⊂→˜Σk and hence strict (k−1)-convexity is precisely what the general Harvey-Lawson theory requires since →Σ∘k=Σ∘k.
Consider Ω⊂RN a bounded domain with C2 boundary with principal curvatures {κi(y)}N−1i=1, unit inner normal ν(y) and tangent space T(y) at each y∈∂Ω. Denote the distance function to the boundary by
d(x):=dist(x,∂Ω), x∈RN. | (2.55) |
Following section 14.6 of [10], will recall some known facts concerning the calculation of κi(y0) at a fixed boundary point y0 and the notion of a principal coordinate system near y0 which yields nice formula for the Hessian of d in suitable tubular neighborhoods of the boundary.
With y0∈∂Ω fixed, choose coordinates x=(x′,xN)∈RN−1×R=RN such that the inner unit normal is ν(y0)=(0,1). Then there exists an open neighborhood N0 of y0 and a function
ϕ:N0∩T(y0)→R of class C2 with Dϕ(y′0)=0 | (2.56) |
so that
∂Ω∩N(y0)={(x′,ϕ(x′)): x′∈N0∩T(y0)} | (2.57) |
and
the principal curvatures {κi(y0)}N−1i=1 are the eigenvalues of D2ϕ(y′0). | (2.58) |
In a principal coordinate system at y0, where one takes the axes x1,…,xN−1 along the associated eigenvectors for D2ϕ(y′0), one has
D2ϕ(y′0)=diag[κ1(y0),…,κN−1(y0)]. | (2.59) |
The following properties of bounded C2 domains are well known and will be used repeatedly in the sequel. For the proofs, see Lemma 14.16 and Lemma 14.17 of [10].
Lemma 2.12. Let Ω⊂RN be a bounded domain with C2 boundary. Then there exists δ>0 such that:
(a) ∂Ω satisfies a uniform interior (and uniform exterior) sphere condition with balls of radius bounded below by δ so that the principal curvatures satisfy for each i∈{1,…,N−1}
|κi(y)|≤1δ foreach y∈∂Ω; | (2.60) |
(b) the distance function d(⋅):=dist(⋅,∂Ω) satisfies
d∈C2(¯Ωδ) | (2.61) |
and
|Dd(x)|=1foreachx∈Ωδ, | (2.62) |
where
Ωδ:={x∈Ω, 0<d(x)<δ}; | (2.63) |
(c) for each x∈Ωδ
thereexistsauniquey=y(x)∈∂Ωsuchthatd(x)=|x−y|; | (2.64) |
(d) from (2.62) one has that D2d(x0) has a zero eigenvalue associated to the eigenvector Dd(x0) for each x0∈Ωδ and using a principal coordinate system based at the point y0=y(x0), which realizes the distance from x0 to the boundary, one has Dd(x0)=(0,…,0,1) and
D2d(x0)=diag[−κ11−κ1d,…,−κN−11−κN−1d,0], | (2.65) |
where κi=κi(y0), d=d(x0) and 1−κid>0 since d<δ and κi satisfies (2.60).
Managing Sk is facilitated by using the principal coordinate systems near the boundary discussed above. Also radial functions are often handy for comparison arguments used in Hopf-type boundary estimates and Hölder regularity arguments, as we will see. In this subsection, we record two lemmas for future use.
Lemma 2.13. Let Ω⊂RN be a bounded domain with C2 boundary. For any g∈C2((0,δ)) and any x0∈Ωδ one has the following formula for the composition v=g∘d and for each j=1,…,N
Sj(D2v(x0))=σj(−κ11−κ1dg′(d),…,−κN−11−κN−1dg′(d),g′′(d)), | (2.66) |
where again κi=κi(y0) and d=d(x0) in a principal coordinate system based at y0∈∂Ω which realizes the distance to x0∈Ωδ as in Lemma 2.12.
Proof. For g∈C2 the Hessian of the composition v=g∘d in Ωδ is given by
D2v=g′(d)D2d+g′′(d)Dd⊗Dd | (2.67) |
which has eigenvalues λN(D2v)=g′′(d) and λi(D2v)=g′(d)ei(d) where {ei(d)}N−1i=1 are the first N−1 eigenvalues of D2d whose expression at x0∈Ωδ in a principal coordinate system based at y0=y(x0) is given by (2.65) and hence
D2v(x0)=diag[−κ11−κ1dg′(d),…,−κN−11−κN−1dg′(d),g′′(d)], | (2.68) |
from which (2.66) follows by the definition of Sj.
Lemma 2.14. For radial functions w(x)=h(|x−x0|) with h∈C2, the eigenvalues of D2w in any punctured neighborhood of x0 are §
§Note that h′′(r)=h′(r)/r is possible; for example, if h(r)=r2. In that case, there is only one distinct eigenvalue with multiplicity N.
h′′(r)withmultiplicityoneandh′(r)/rwithmultiplicityN−1, | (2.69) |
where r:=|x−x0| and hence ¶
¶Here and below, (nk):=n!k!(n−k)! for integers satisfying n≥k≥0.
Sk(D2w(x))=h′′(r)(h′(r)r)k−1(N−1k−1)+(h′(r)r)k(N−1k); | (2.70) |
that is,
Sk(D2w(x))=(h′(r)r)k−1(N−1k−1)[h′′(r)+h′(r)rN−kk]. | (2.71) |
Proof. The claim in (2.69) is well known, from which (2.70) and (2.71) follow easily.
As suggested in the title, we will make use of various comparison and maximum principles for Σk-admissible viscosity subsolutions and supersolution in the sense of Definition 2.5 and the subsequent remarks and examples. While they will be special cases of the results in [6,7,11], for the convenience of the reader we will give the precise statements and some indication of the proofs. In all that follows Ω will be an open bounded domain in RN.
We begin the most basic comparison result, which concerns a Σk-subharmonic and Σk-superharmonic pair, as presented in Example 2.6.
Theorem 3.1. Suppose that u∈USC(¯Ω) and v∈LSC(¯Ω) are a Σk-admissible viscosity subsolution/supersolution pair for the homogeneous equation Sk(D2u)=0 in Ω. Then the comparison principle holds; that is,
u≤v on ∂Ω ⇒ u≤v on Ω. | (3.1) |
Proof. The hypothesis is equivalent to saying that u and v are Σk-subharmonic and Σk-superharmonic in Ω, as discussed in Example 2.6. Since Σk is a pure second order subequation, one has the comparison principle (3.1) as a corollary of the comparison principle of [11] (see also Theorem 9.3 of [7]). The main ingredients in the proof are that −v is ˜Σk-subharmonic and that w:=u−v is ˜P-subharmonic (coming from the P-monotonicity of Σk and its dual), for which the zero maximum principle holds
w≤0 on ∂Ω ⇒ w≤0 on Ω. | (3.2) |
See section 7 of [7] for details in the general case of ˜M-monotone subequations.
As noted in (2.20), the admissibility constraint sets satisfy
Σk⊂Σ1=H for each k=1,…,N |
and hence each u which is Σk-subharmonic on Ω will be H-subharmonic on Ω and hence u satisfies the mean value inequality
u(x0)≤1|Br(x0)|∫Br(x0)u(x)dx for each Br(x0)⊂Ω. | (3.3) |
An immediate consequence of (3.3) is the strong maximum principle.
Theorem 3.2. For each u∈USC(Ω) which is Σk-subharmonic (k-convex) on a bounded domain (open, connected set) one has
ifthereexistsx0∈Ωwithu(x0)=supΩu,thenuisconstantinΩ. | (3.4) |
In particular, if u∈USC(¯Ω) is Σk-subharmonic (k-convex) in Ω then
u≤0 on ∂Ω ⇒ u<0 in Ω or u≡0 in Ω. | (3.5) |
Remark 3.3. Notice that (3.5) is the strong form of the zero maximum principle
u≤0 on ∂Ω ⇒ u≤0 on Ω. | (3.6) |
The weak form (3.6) of the zero maximum principle is also a simple corollary the comparison principle in Theorem 3.1. Indeed, one compares the Σk-subarmonic u with the smooth function v≡0. Since ~Σk also satisfies the positivity property (2.27), the coherence property of Remark 2.4 holds and hence v≡0 is Σk-superharmonic since 0=Sk(−v)∈~Σk.
We conclude this section with a comparison result which is tailored for some of the pointwise estimates we will need.
Theorem 3.4. Let c≥0 be fixed. Suppose that u∈LSC(¯Ω) satisfies ‖
‖By this we mean that u is a Σk-admissible viscosity supersolution of the equation Sk(D2u)−c=0.
Sk(D2u)≤c in Ω. | (3.7) |
Suppose that v∈C2(Ω)∩C(¯Ω) is a strictly k-convex strict subsolution, that is,
D2v(x)∈Σ∘k and Sk(D2v(x))>c forall x∈Ω. | (3.8) |
Then, one has the comparison principle
v≤u on ∂Ω ⇒ v≤u on Ω. | (3.9) |
Proof. Suppose not, then v−u∈USC(¯Ω) will have a (positive) maximum at some interior point x0∈Ω. Hence u−v will have a (negative) minimum at x0. Choose φ=v in Definition 2.5 (b) of a Σk-admissible supersolution u to find
Sk(D2v(x0))≤c or D2v(x)∉Σ∘k, |
which contradicts (3.8).
Some variants of these principles will also be present in some of the proofs.
For the minimum principle characterization of Theorem 5.4 and for the global Hölder regularity result of Theorem 6.3, we will make use of various barrier functions which provide some needed one-sided bounds near the boundary ∂Ω of bounded C2 domains. The arguments are standard, but the details involve having a sufficiently robust calculus for the k-Hessian.
The first estimate is a form of the Hopf lemma which will be applied to the subsolutions ψ competing in the Definition of the principal eigenvalue when we prove the minimum principle characterization of Theorem 5.4.
Proposition 4.1. Given λ≥0. Suppose that ψ∈USC(¯Ω) is a k-convex subsolution i.e., a Σk-admissible subsolution in the sense of Definition 2.5 of
{Sk(D2ψ)+λψ|ψ|k−1=0in Ωψ=0on ∂Ω | (4.1) |
which is negative on Ω. Then
there exists C1>0 such that ψ(x)≤−C1d(x) for all x∈Ωδ/2, | (4.2) |
where Ωδ/2 is the tubular neighborhood defined as in (2.63) with δ>0 such that the conclusions of Lemma 2.12 hold in the larger neighborhood Ωδ.
Proof. One uses the usual argument of comparing ψ with a smooth radial function w which is a strict supersolution of (4.1) and dominates ψ on the boundary of an annular region within the good tubular neighborhood Ωδ (see Proposition 2.5 of [4], for example). For completeness, we give the argument.
For each x0∈Ωδ/2, denote by y0=y0(x0)∈∂Ω which realizes the distance to the boundary and by z0=z0(x0)=y0+δ(x0−y0)/|x0−y0| the center of an interior ball such that
Bδ(z0)⊂Ωand¯Bδ(z0)∩(RN∖Ω)={y0}. | (4.3) |
Consider the smooth negative radially symmetric function
w(x)=C0(e−mδ−e−m|x−z0|) for each x∈A:=Bδ(z0)∖¯Bδ/2(z0), | (4.4) |
where C0 and m are chosen to satisfy
C0≤supΩ∖Ωδ/2ψe−mδ−e−mδ/2 | (4.5) |
and
m>2(N−k)kδ. | (4.6) |
One has w(x)≥ψ(x) on ∂A; which is trivial on the outer boundary where w vanishes and the choice of C0 in (4.5) is used on the inner boundary which is contained in Ω∖Ωδ/2. The key point is to show that
w(x)≥ψ(x) on A. | (4.7) |
Suppose, by contradiction, that ψ−w has a positive maximum point at ˉx in the interior of A. By Definition 2.5(a) of the Σk-admissible subsolution ψ of the PDE, one has
Sk(D2w(ˉx))≥−λψ(ˉx)|ψ(ˉx)|k−1≥0 |
But, using the radial calculation (2.71) with h(r)=C0(e−mδ−e−mr) and r=x−z0, with the choice on m in (4.6) where r>δ/2, one has
Sk(D2w(x))=Ck0mke−mkrrk(N−1k−1)[−mr+N−kk]<0 |
which yields a contradiction. Hence, for x0∈Ωδ/2 one has
ψ(x0)≤w(x0)=C0e−mδ(1−emd(x0))≤C0e−mδ(−md(x0)) |
and one can take C1=−mC0e−mδ.
The next estimate gives a lower bound near the boundary for the supersolutions treated in the minimum principle characterization of Theorem 5.4.
Proposition 4.2. Let Ω be a bounded strictly (k−1)-convex domain and λ≥0. Suppose that u∈LSC(¯Ω) is a Σk-admissible supersolution ** of
**By this we mean that u is a Σk-admissible supersolution of Sk(D2u)+λu|u|k−1=0 in Ω in the sense of Definition 2.5(b) and that u≥0 on ∂Ω.
{Sk(D2u)+λu|u|k−1=0in Ωu=0on ∂Ω. | (4.8) |
Then
there exist C2>0 and d0>0 such that u(x)≥−C2d(x) for all x∈Ωd0, | (4.9) |
where, as always, Ωd0:={x∈Ω: 0<d(x)<d0}.
Before giving the proof, a pair of remarks are in order. Since u(x)≥0=−C2d(x) for each x∈∂Ω, the lower bound in (4.9) holds trivially there. Moreover, having u>0 at boundary points facilitates having a negative lower bound and the interesting case concerns u≡0 on ∂Ω.
Proof of Proposition 4.2: Since ∂Ω∈C2, by Lemma 2.12, there exists δ>0 such that d∈C2(¯Ωδ). Consider the comparison function φ∈C2(¯Ωδ) defined by
φ(x):=e−td(x)−1:=g(d(x)) with t>0 sufficiently large. | (4.10) |
Claim: For t sufficiently large and d0 sufficiently small, one has:
D2φ(x0)∈Σ∘k foreach x0∈Ωd0; | (4.11) |
φ≤u on ∂Ωd0; | (4.12) |
Sk(D2φ)+λφ|φ|k−1>0 in Ωd0 | (4.13) |
That is, on a sufficiently small tubular domain Ωd0, the function φ is a C2 strictly k-convex strict subsolution of the eigenvalue equation which is dominated by the supersolution u on ∂Ωd0.
Given the claim, the Σk-admissible supersolution u must satisfy
u≥φ on Ωd0. | (4.14) |
Indeed, if (4.14) were to be false, then u−φ∈LSC(¯Ωd0) would have its (negative) minimum at some x0∈Ωd0 (using (4.12)). By Definition 2.5(b), one would then have
Sk(D2φ(x0))+λφ(x0)|φ(x0)|k−1≤0 or D2φ(x0)∉Σ∘k, |
which contradicts (4.13)–(4.11). The relation (4.14) gives the barrier estimate (4.9) since
u(x)≥φ(x):=e−td(x)−1≥−C2d(x), x∈Ωd0 |
if C2 is chosen to satisfy C2≥t.
Thus it remains only to verify the Claim. We begin with the strict k-convexity of φ claimed in (4.11). Using Lemma 2.13 on φ=g∘d and calculating the needed derivatives of g(d):=e−td−1, on Ωδ one has for each j=1,…,k
Sj(D2φ(x0))=tje−jtd(x0)σj(κ1(y0)1−κ1(y0)d(x0),…,κN−1(y0)1−κN−1(y0)d(x0),t), | (4.15) |
where again we use a principal coordinate system based at y0=y(x0)∈∂Ω which realizes the distance to x0∈Ωδ. Now, using the strict (k−1)-convexity of ∂Ω, by Lemma 2.10 there exists R>0 and β0>0 so that
σj(κ1(y0),…,κN−1(y0),R)≥β0>0 for each y0∈∂Ω and each j=1,…k. | (4.16) |
We have used the continuity of each σj and the compactness of ∂Ω to pick up the positive lower bound β0. Since each σj (with 1≤j≤k) is increasing in Γk with respect to λN, we can freely replace R by any t≥R in (4.16). Again by continuity and compactness, we can choose d0≤δ so that for each x0∈Ωd0 and for each j=1,…k, one has
σj(κ1(y0)1−κ1(y0)d(x0),…,κN−1(y0)1−κN−1(y0)d(x0),R)≥β02>0. | (4.17) |
Indeed, with
μ:=maxy0∈∂Ωmax1≤i≤N−1|κi(y0)| | (4.18) |
p:=(κ1(y0)1−κ1(y0)d(x0),…,κN−1(y0)1−κN−1(y0)d(x0)) and q:=(κ1(y0),…,κN−1(y0)), |
by choosing d0≤1/(2μ), for every x0∈¯Ωd0 one has
|p−q|≤2√N−1μd0, |
which can be made as small as needed to ensure that (4.17) follows from (4.16) (by taking d0 even smaller if needed).
By choosing t:=R in (4.15) and using (4.17), one has
Sj(D2φ(x0))≥Rje−jRd0β02>0 for each x0∈Ωd0 and j=1,…k, | (4.19) |
and hence the strict k-convexity of φ on Ωd0.
Next we verify the claim (4.12) concerning boundary values. The claim is trivial on the outer boundary ∂Ω where φ vanishes and u≥0. On the compact inner boundary (where d(x)=d0), by reducing d0 if need be, we can assume that u≥−1/2 (since u is lower semi-continuous and is non-negative on ∂Ω). Hence it suffices to choose t>0 large enough so that
φ=e−td0−1≤−1/2≤u. |
Recall that we may freely increase t≥R without spoiling the k-convexity of φ as noted above.
Finally, we need to verify the subsolution claim (4.13) which using the negativity of φ is equivalent to
Sk(D2φ(x0))>λ|φ(x0)|k for each x0∈Ωd0. | (4.20) |
Using (4.15) and (4.17) with j=k, for each x0∈Ωd0, we have
Sk(D2φ(x0))=tke−ktd(x0)σk(κ1(y0)1−κ1(y0)d(x0),…,κN−1(y0)1−κN−1(y0)d(x0),t)>tke−ktd(x0)β02. | (4.21) |
Now, on Ωd0 (where d(x0)<d0), we have
λ|φ(x0)|k=λ(1−e−td(x0))k<λ(1−e−td0)k. | (4.22) |
Combining (4.21) with (4.22), we will have (4.20) provided that
tke−ktd(x0)β02>λ(1−e−td0)k, |
which holds if d0 is chosen small enough.
The final boundary estimate we will need is similar to the preceding estimate and will be employed in the proof of the uniform Hölder regularity for a sequence of solutions tending to a principal eigenfunction, see Theorems 6.3 and 6.6.
Proposition 4.3. Let Ω be a bounded strictly (k−1)-convex domain and let f≥0 be a bounded continuous function on Ω. Suppose that u∈LSC(¯Ω) is a Σk-admissible supersolution of
{Sk(D2u)=fin Ωu=0on ∂Ω | (4.23) |
Then there exist d0>0 and C3>0 such that
u(x)≥−C3d(x) for all x∈Ωd0, | (4.24) |
where, as always, Ωd0:={x∈Ω: 0<d(x)<d0}.
Proof. Consider the family of comparison functions used in Proposition 4.2; that is,
φ(x)=e−td(x)−1 with t>0 sufficiently large. | (4.25) |
We have seen in (4.19) that there exists d0>0 sufficiently small such that
D2φ(x0)∈Σ∘k for each x0∈Ωd0. | (4.26) |
In particular, if we choose t=1/d0 in (4.21) (so that t is large if d0 is small) we have
Sk(φ(x0))>d−k0e−kβ02>supΩf for each x0∈Ωd0, | (4.27) |
provided that d0 is chosen small enough. Using (4.26) and (4.27), for d0 sufficiently small, we have that φ(⋅)=e−d(⋅)/d0−1∈C2(Ωd0)∩C(¯Ωd0) is a strictly k-convex strict subsolution of the equation
Sk(D2u)=c:=supΩf in Ωd0. | (4.28) |
By hypothesis, u∈LSC(¯Ω) is also a Σk-admissible supersolution of (4.28) and hence by the comparison principle of Theorem 3.4 we will have
u(x)≥φ(x)=e−d(x)/d0−1 for all x∈Ωd0, | (4.29) |
provided that this inequality holds on the boundary; that is,
u(x)≥φ(x)=e−d(x)/d0−1 for all x∈∂Ωd0. | (4.30) |
For the boundary inequality (4.30), on the outer boundary ∂Ω we have u≥0=φ since u is a supersolution of (4.23) and d(x)=0 on ∂Ω. On the inner boundary ∂Ωd0∩Ω (where d(x)=d0) we have
φ(x)=e−1−1:=−ε<0. | (4.31) |
Since u∈LSC(¯Ω) with u≥0 on ∂Ω there exits δ=δ(ε)>0 such
u(x)>−ε for each x∈Ω such that dist(x,∂Ω)<δ. | (4.32) |
Choosing d0 even smaller so that d0≤δ in (4.32) then shows that (4.30) holds on the inner boundary ∂Ωd0∩Ω as well.
Finally, the comparison estimate (4.29) gives the barrier estimate (4.24) for C3>1/d0.
In all that follows, Ω⊂RN will be a bounded open domain with C2 boundary which is strictly (k−1)-convex in the sense of Definition 2.9. Denote by
Φ−k(Ω):={ψ∈USC(Ω):ψ is k-convex and negative in Ω}, | (5.1) |
where the notion of k-convexity is that of Definition 2.1. Notice that since ψ is bounded from above (by zero) on Ω one can extend ψ to a USC(¯Ω) function in a canonical way by letting
ψ(x0):=lim supx→x0x∈Ωu(x), for each x0∈∂Ω. | (5.2) |
In particular, since ψ<0 on Ω this extension also satisfies
ψ≤0 on ∂Ω. | (5.3) |
We will freely make use of this extension so that Proposition 4.1 (Hopf's Lemma) applies to give the boundary estimate (4.2) for the canonical extension of ψ∈Φ−k(Ω).
The following definition gives a candidate for the principal eigenvalue associated to a negative k-convex eigenfunction of the k-Hessian Sk(D2u)=σk(λ(D2u)).
Definition 5.1. For each k∈{1,…,N} fixed, define
λ−1(Sk,Σk):=supΛk | (5.4) |
where
Λk:={λ∈R: ∃ψ∈Φ−k(Ω) with Sk(D2ψ)+λψ|ψ|k−1≥0 in Ω}. | (5.5) |
The meaning of the differential inequality in (5.5) is in the viscosity sense; that is, for each x0∈Ω and for each φ which is C2 near x0 one has that
ψ−φ has a local maximum in x0 ⇒ Sk(D2φ(x0))+λψ(x0)|ψ(x0)|k−1≥0. | (5.6) |
Moreover, since ψ∈Φ−k(Ω) is k-convex, ψ is a Σk-admissible subsolution of the PDE in the sense of Definition 2.5 (a), whose canonical extension to the boundary (5.2) is admissible for Proposition 4.1 (as noted above).
About the definition, a few elementary remarks are in order which we record in the following lemma.
Lemma 5.2. Let λ−1(Sk,Σk) and Λk be as in Definition 5.1. Then the following facts hold.
(a) (−∞,λ−1(Sk,Σk))⊂Λk, or equivalently, if λ<λ−1(Sk,Σk) then there exists ψ∈USC(Ω) which is k-convex and negative in Ω and satisfies
Sk(D2ψ)+λψ|ψ|k−1≥0 in Ω. | (5.7) |
(b) If Ω (which is bounded) is contained in BR(0) then one has the estimate
λ−1(Sk,Σk)≥2kCN,kR−2kwhere CN,k=(Nk). | (5.8) |
In particular, λ−1(Sk,Σk) is positive.
Proof. For the part (a), we first claim that if λ∈Λk then (−∞,λ]⊂Λk. By the definition of Λk, there is ψ∈Φ−k(Ω) as defined in (5.1) which satisfies (5.7). If λ∗<λ then this ψ∈Φ−k(Ω) satisfies (5.7) with λ∗ in place of λ. Indeed, for each x0∈Ω and each φ which is C2 near x0 one has (5.6) and hence
Sk(D2φ(x0))+λ∗ψ(x0)|ψ(x0)|k−1+(λ−λ∗)ψ(x0)|ψ(x0)|k−1≥0, |
but the last term is negative and hence the claim.
Now, if λ<λ−1(Sk,Σk) then by the definition of λ−1(Sk,Σk) there must exist λ∗=λ+ε∗ between λ and λ−1(Sk,Σk) which belongs to Λk and hence λ+ε∈Λk for each ε∈[0,ε∗] by the claim proved above, which completes the proof of part (a).
For part (b), consider the convex (and hence k-convex) function ψ(x):=|x|2−R2 which is negative on Ω⊂BR(0). One has D2ψ(x)=2I for each x∈Ω and hence
Sk(D2ψ(x))+λψ(x)|ψ(x)|k−1=2kCN,k−λ(R2−|x|2)k≥2kCN,k−λR2k≥0, |
provided λR2k≤2kCN,k. The claim (5.8) follows.
Remark 5.3. While the lower bound (5.8) shows that λ−1(Sk,Σk) is positive, in Theorem 7.1 we will also give an upper bound for λ−1(Sk,Σk) on domains which contain some ball BR(0) which shows that λ−1(Sk,Σk)<+∞.
We are now ready for the main result of this section, which we state in the nonlinear case i.e., k>1.
Theorem 5.4. Let k∈{2,…,N} and let Ω be a strictly (k−1)-convex domain in RN. For every λ<λ−1(Sk,Σk) and for every u∈LSC(¯Ω) which is Σk-admissible supersolution of
Sk(D2u)+λu|u|k−1=0 in Ω, | (5.9) |
one has the following minimum principle
u≥0 on ∂Ω ⇒ u≥0 in Ω. | (5.10) |
Before giving the proof, a pair of remarks are in order.
Remark 5.5. If λ≤0, then the gradient-free equation (5.9) is proper elliptic on the constraint set R×Σk and the maximum/minimum principle for (R×Σk)-admissible viscosity subsolutions/supersolutions of (5.9) follows from [7] (see section 11.1). Hence, we will restrict attention to the interesting case
0<λ<λ−1(Sk,Σk). | (5.11) |
Proof of Theorem 5.4. We argue by contradiction. Assume that
there exists x∈Ω such that u(x)<0 | (5.12) |
and so u∈LSC(¯Ω) will have a negative minimum on ¯Ω in some interior point ˉx∈Ω. Let ψ≤0 on ∂Ω be a Σk-admissible subsolution of
Sk(D2ψ)+˜λψ|ψ|k−1≥0 in Ω for a fixed ˜λ∈(λ,λ−1(Sk,Σk)). | (5.13) |
We will compare u with γψ where
γ∈(0,γ′:=supΩuψ) | (5.14) |
is to be suitably chosen and ψ∈USC(¯Ω) is k-convex, negative in Ω. Notice that such values of ˜λ>0 exist by (5.11) and that such a ψ exists by Lemma 5.2 (a), where we take the canonical USC extension to the boundary of (5.2) so that (5.3) also holds for ψ.
Notice also that if ψ solves (5.13) then so does γψ. Indeed, for each x0∈Ω, if γψ−φ has a local max in x0 then ψ−1γφ with γ>0 does too and hence
γ−kSk(D2φ(x0))+˜λψ(x0)|ψ(x0)|k−1≥0, | (5.15) |
which gives (5.13) for \gamma \psi by multiplying (5.15) by \gamma^k > 0 .
Step 1: Show that \gamma' > 0 defined in (5.14) is finite: that is, one has
\begin{equation} \sup\limits_{\Omega} \frac{u}{\psi} \lt + \infty. \end{equation} | (5.16) |
We begin by noting that \psi < 0 on \Omega and we have assumed that u has a negative minimum at \bar{x} \in \Omega so that the ratio is positive in \bar{x} . Near the boundary, we make use of the boundary estimates of Proposition 4.1 for \psi and Proposition 4.2 for u to say that there exist C_1, C_2 > 0 such that
\begin{equation} \psi(x)\leq -C_1 \, d(x) \ \text{for all} \ x \in \Omega_{\delta/2} \end{equation} | (5.17) |
and
\begin{equation} u(x) \geq -C_2 \, d(x) \ \text{for all} \ x \in \Omega_{d_0}, \end{equation} | (5.18) |
where \delta > 0 is the parameter of Lemma 2.12 defining a good tubular neighborhood of \partial \Omega \in C^2 and d_0 \leq \delta depends on \mu (as defined in (4.18), which bounds the absolute values of the principal curvatures of \partial \Omega ), the monotonicity properties of \sigma_j for j \leq k on \overline{\Gamma}_k and their moduli of continuity. Hence, by picking
\rho \leq \min \{ d_0, \delta/2 \} |
and recalling that -\psi > 0 on \Omega , we can use both (5.17) and (5.18) on \Omega_{\rho} to find
\frac{u(x)}{\psi(x)} = \frac{-u(x)}{-\psi(x)} \leq \frac{C_2 \, d(x)}{C_1 \, d(x)} = \frac{C_2}{C_1} \ \ \text{for all} \ x \in \Omega_{\rho}; |
that is
\begin{equation} \sup\limits_{\Omega_{\rho}} \frac{u}{\psi} \leq \frac{C_2}{C_1} \lt +\infty. \end{equation} | (5.19) |
Now, on the compact set K : = \overline{\Omega \setminus \Omega_{\rho}} where -\psi \in \mathrm{LSC}(K) and positive and -u \in \mathrm{USC}(K) one has the existence of \widetilde{C}_1 > 0 and \widetilde{C}_2 such that
-\psi(x) \geq \widetilde{C}_1 \gt 0 \ \ \text{and} \ \ -u(x) \leq \widetilde{C}_2 \ \ \text{for each} \ x \in K |
to find
\begin{equation} \sup\limits_{K} \frac{u}{\psi} \leq \frac{\widetilde{C}_2}{\widetilde{C}_1} \lt +\infty. \end{equation} | (5.20) |
Combining (5.19) with (5.20) gives the needed (5.16).
Step 2: Reduce the proof to showing that there exists \widetilde{x} \in \Omega such that u(\widetilde{x}) < 0 and
\begin{equation} \lambda |u(\widetilde{x})|^k \geq \gamma^k \widetilde{\lambda} |\psi(\widetilde{x})|^k. \end{equation} | (5.21) |
Indeed, recalling that u(\widetilde{x}), \psi(\widetilde{x}) < 0 , 0 < \lambda < \widetilde{\lambda} < \lambda_1^{-} and \gamma^{\prime} : = \sup_{\Omega}(u/\psi) , from (5.21) one finds
\begin{equation} \frac{\widetilde{\lambda}}{\lambda}\gamma^k \leq \left( \frac{ u(\widetilde{x})}{\psi(\widetilde{x})} \right)^k \leq (\gamma')^k. \end{equation} | (5.22) |
Now, choose the free parameter \gamma \in (0, \gamma^{\prime}) to satisfy
\begin{equation} \gamma \gt \left( \frac{\lambda}{\widetilde{\lambda}} \right)^{1/k} \gamma^{\prime}, \end{equation} | (5.23) |
which can be done since \gamma \in (0, \gamma^{\prime}) and (\lambda/\widetilde{\lambda})^{1/k} < 1 . Raising the inequality (5.23) to power k and multiplying by \widetilde{\lambda} > 0 gives a contradiction to the inequality (5.22). This completes the proof of the theorem, modulo showing that such an \widetilde{x} exists.
Step 3: Exhibit \widetilde{x} \in \Omega such that u(\widetilde{x}) < 0 and (5.21) holds.
In order to find \tilde{x} \in \Omega such that (5.21) holds when comparing u to \gamma \psi , we make use of the classical viscosity device of looking at the maximum values of the family of upper semicontinuous functions defined by doubling variables and with an increasing (in j \in {\mathbb N} ) quadratic penalization
\begin{equation} \Psi_j(x,y): = \gamma \psi(x) - u(y) - \frac{j}{2}|x-y|^2, \ \ (x,y) \in \overline{\Omega} \times \overline{\Omega}, \ j \in {\mathbb N}. \end{equation} | (5.24) |
For simplicity of notation, we will suppress the free parameter \gamma \in (0, \gamma^{\prime}) in the notation for \Psi_j (as well in certain \gamma -dependent quantities below), thinking of \gamma \in (0, \gamma^{\prime}) as arbitrary, but fixed.
First, notice that for each j \in {\mathbb N} , \Psi_j \in \mathrm{USC}(\overline{\Omega} \times \overline{\Omega}) will have a maximum value
\begin{equation} M_j : = \max\limits_{(x,y) \in \overline{\Omega} \times \overline{\Omega} } \Psi_j(x,y) \lt +\infty. \end{equation} | (5.25) |
Claim: For each j \in {\mathbb N} , the maximum M_j defined in (5.25) is positive.
For each fixed \gamma \in (0, \gamma^{\prime}) , we will show that
\begin{equation} \Psi_j (x,x) : = \gamma \psi(x) - u(x). \end{equation} | (5.26) |
must have a positive value in the interior of \Omega , and hence the claim. Assume to the contrary that \Psi_j (x, x) \leq 0 on \Omega ; that is, for each x \in \Omega , assume that
\gamma \psi(x) \leq u(x) \ \ \Leftrightarrow \ \ \frac{u(x)}{\psi(x)} \leq \gamma, |
since \psi < 0 on \Omega . This implies that \gamma^{\prime} which is the sup of u/\psi satisfies \gamma^{\prime}\leq\gamma . However this is a contradiction since \gamma < \gamma^{\prime} , which completes the claim.
Hence, using the claim, for each fixed \gamma \in (0, \gamma^{\prime}) , there exists \bar{x} \in \Omega such that
\begin{equation} M_j: = \max\limits_{\overline{\Omega} \times \overline{\Omega}} \Psi_j \geq \Psi_j(\bar{x}, \bar{x}) = \gamma \psi(\bar{x}) - u(\bar{x}) : = \bar{m} \gt 0, \ \forall \, j \in {\mathbb N}. \end{equation} | (5.27) |
Notice that the maximum values M_j decrease as j increases so that
\begin{equation} M_{\infty} : = \lim\limits_{j \to + \infty} M_j = \inf\limits_{j \in {\mathbb N}} M_j \geq \bar{m} \gt 0. \end{equation} | (5.28) |
Using the finiteness of the limit M_{\infty} and the fact that u \geq 0 and \psi \leq 0 on \partial \Omega one has the following standard facts (see, for example, Lemma 3.1 of [8]).
Lemma 5.6. For each j \in {\mathbb N} consider any pair (x_j, y_j) \in \overline{\Omega} \times \overline{\Omega} such that
\begin{equation} M_j : = \max\limits_{\overline{\Omega} \times \overline{\Omega}} \Psi_j = \Psi_j(x_j, y_j). \end{equation} | (5.29) |
One has
\begin{equation} \lim\limits_{j \to +\infty} j |x_j - y_j|^2 = 0 \ \ and\ \ hence \ \ (x_j - y_j) \to 0 \ \text{as} \ j \to +\infty; \end{equation} | (5.30) |
\begin{equation} (x_j, y_j) \in \Omega \times \Omega \ \ for\ \ all \ j \ sufficiently\ \ large; \end{equation} | (5.31) |
and for any accumulation point \tilde{x} of the bounded sequence \{x_j\}_{j \in {\mathbb N}} one has
\begin{equation} 0 \lt \bar{m} \leq M_{\infty} = \gamma \psi(\tilde{x}) - u(\tilde{x}) = \max\limits_{(x,y) \in \overline{\Omega} \times \overline{\Omega}} (\gamma \psi(x) - u(y)) \end{equation} | (5.32) |
and hence \tilde{x} \in \Omega since \gamma \psi, -u \leq 0 on \partial \Omega .
Proof. For completeness, we sketch the argument. The claim (5.30) follows from the fact that for each j \in {\mathbb N} one has
M_{j/2} \geq \Psi_{j/2}(x_j,y_j) = M_j + \frac{j}{4} |x_j - y_j|^2 |
and hence by (5.28)
0 \leq \frac{j}{4} |x_j - y_j|^2 \leq M_{j/2} - M_j \to 0 |
Next, by (5.29) with \Psi_j as defined in (5.24), one has
M_j = \gamma \psi(x_j) - u(y_j) - \frac{j}{2} |x_j - y_j|^2 \geq \bar{m} \gt 0 |
and hence (5.30) yields
\gamma \psi(x_j) - u(y_j) \gt 0 \ \ \text{for all sufficiently large} \ j. |
Hence for large j one has (5.31) since \gamma \psi, -u \leq 0 on \partial \Omega and x_j - y_j \to 0 as j \to + \infty . Finally, for the claim (5.32), if x_{j_k} \to \tilde{x} as k \to + \infty , then so does y_{j_k} and using (5.30) plus the fact that \gamma \psi, -u \in \mathrm{USC}(\overline{\Omega}) yields
M_{\infty} = \limsup\limits_{k \to + \infty} \left(\gamma \psi (x_{j_k}) - u(y_{j_k}\right) \leq \gamma \psi(\tilde{x}) - u(\tilde{x}) = \Psi_{j_k}(\tilde{x}, \tilde{x}) \leq M_{j_k}, \ \ \forall \, k \in {\mathbb N}. |
One can now exhibit \widetilde{x} \in \Omega such that u(\widetilde{x}) < 0 and (5.21) holds. The idea is to apply Ishii's lemma (as given in the discussion of the formulas (3.9) and (3.10) in Crandall-Ishii-Lions [8]) along positive interior (local) maximum points of \Psi_j and using that \gamma \psi and u are viscosity sub and supersolutions in \Omega . More precisely, if
\Psi_j(x,y) : = \gamma \psi(x) - u(y) - \frac{j}{2} |x-y|^2 \in \mathrm{USC}(\overline{\Omega} \times \overline{\Omega}) |
has a local maximum in (x_j, y_j) \in \Omega \times \Omega , then by Lemma 5.6, for large j these local maxima lie in \Omega \times \Omega and by Ishii's lemma there exist X_j, Y_j \in \mathcal{S}(N) such that
\begin{equation} (j(x_j - y_j), X_j) \in {\overline{J}}^{2,+} \gamma \psi(x_j) \ \ \text{and} \ \ (j (x_j - y_j), Y_j) \in {\overline{J}}^{2,-} u(y_j) \end{equation} | (5.33) |
where
\begin{equation} X_j \leq Y_j \ \ \text{in} \ \mathcal{S}(N). \end{equation} | (5.34) |
Furthermore, by the last part of Lemma 5.6, there exists \widetilde{x} \in \Omega such that, up to a subsequence,
\begin{equation} (x_j, y_j) \to \widetilde{x} \ \ \text{as} \ j \to +\infty. \end{equation} | (5.35) |
Now, since \gamma \psi is \Sigma_k -subharmonic ( k -convex) in \Omega , for each x \in \Omega and for every p \in {\mathbb R}^N one has
\begin{equation} (p,A) \in J^{2,+} \gamma \psi(x) \ \ \Rightarrow A \in \Sigma_k, \end{equation} | (5.36) |
but \Sigma_k is closed and from the first statement of (5.33) it follows that
\begin{equation} X_j \in \Sigma_k. \end{equation} | (5.37) |
By the positivity property (2.27), combining (5.34) and (5.37) yields
\begin{equation} Y_j \in \Sigma_k. \end{equation} | (5.38) |
We remark that this is the key observation that indicates why Ishii's lemma continues to be useful in the case of viscosity solutions with admissibility constraints satisfying the positivity property (2.27).
Next, using that \gamma \psi and u are \Sigma_k -admissible subsolutions and supersolutions of (5.13) and (5.9) respectively, one has for all large j
\begin{equation} S_k(X_j) + \widetilde{\lambda} \gamma^k \psi(x_j) |\psi(x_j)|^{k-1} \geq 0 \end{equation} | (5.39) |
and
\begin{equation} S_k(Y_j) + \lambda u(y_j) |u(y_j)|^{k-1} \leq 0 \end{equation} | (5.40) |
where we have used the fact that Y_j \in \Sigma_k in the supersolution definition (see Definition 2.5 (b)). Using (5.39), (5.40) and the monotonicity property (2.29) of S_k on \Sigma_k (which applies by (5.37) and (5.34)), for all large j \in {\mathbb N} we have
\begin{equation} - \widetilde{\lambda} \gamma^k (\psi(x_j) |\psi(x_j)|^{k-1} \leq S_k(X_j) \leq S_k(Y_j) \leq u(y_j) \lambda|u(y_j)|^{k-1}. \end{equation} | (5.41) |
Since \psi < 0 on \Omega , all of the expressions in (5.41) are positive and hence
\begin{equation} u(y_j) \lt 0 \ \ \text{for all large} \ j. \end{equation} | (5.42) |
Now since -\psi > 0 is \mathrm{LSC}(\Omega) and -u \in \mathrm{USC}(\Omega) , from (5.41) one finds
\begin{equation} 0 \lt \widetilde{\lambda}\gamma^k(-\psi(\widetilde{x}))^k \leq \liminf\limits_{j \to +\infty} \leq \widetilde{\lambda} \gamma^k (- \psi (x_j))^k \leq \limsup\limits_{j \to +\infty} \lambda (-u(y_j))^k \leq \lambda(-u(\widetilde{x}))^k, \end{equation} | (5.43) |
which gives the needed inequality (5.21). Finally, since u is \mathrm{LSC}(\Omega) , by (5.42), we have u(\widetilde{x}) \leq 0 , but it cannot vanish by (5.43). Thus u(\widetilde{x}) < 0 as needed.
An immediate consequence of the minimum principle are the following characterizations of the principal eigenvalue of S_k discussed by Wang [32] and Lions [26] (in the case k = N ) using the variational structure of S_k . See also Jacobsen [22] for a bifurcation approach.
Corollary 5.7. Let \Omega be as in Theorem 5.4 and let k \geq 2 . Then \lambda_1^{-}(S_k, \Sigma_k) as defined by (5.4) and (5.5) is equal to \lambda_1^{(k)} defined by
\begin{equation} \lambda_1^{(k)} : = \inf\limits_{u \in \Phi_0^k(\Omega)} \left\{ - \int_{\Omega} u S_k(D^2u) \, dx: \ \ ||u||_{L^{k + 1}(\Omega)} = 1 \right\}, \end{equation} | (5.44) |
where \Phi_0^k(\Omega) = \{ u \in C^2(\Omega): S_k(D^2 u) \in \Gamma_k \ \ {\rm and} \ \ u = 0 \ {\rm on} \ \partial \Omega \} and \Gamma_k is the open cone (2.3). When k = N , one has
\begin{equation} \mbox{ $\lambda_1^{(N)} : = \inf\{ \lambda_1^a: \ a \in C(\overline{\Omega}, \mathcal{S}(N)) \ \text{such that} \ \ a \gt 0 , {\rm det}\, a \geq N^{-N} \ \text{in} \ \overline{\Omega}, \}$} \end{equation} | (5.45) |
and \lambda_1^a is the first eigenvalue of the uniformly elliptic operator - \sum_{i, j = 1}^N a_{ij} D_{ij} .
Proof. Since there exists a k -convex principal eigenfunction \psi_1 which is negative in the interior and vanishes on the boundary, by the definition of \lambda_1^{-}(S_k, \Sigma_k) , one has \lambda_1^{(k)} \leq \lambda_1^{-}(S_k, \Sigma_k) . If \lambda_1^{(k)} < \lambda_1^{-}(S_k, \Sigma_k) , then \psi_1 would be a \Sigma_k -admissible supersolution of (5.9) with \lambda = \lambda_1^{(k)} and hence \psi_1 \geq 0 in \Omega by the minimum principle (5.9), which is absurd.
Even though Corollary 5.7 shows that a negative principal eigenfunction \psi_1 exists for \lambda_1^{-} = \lambda_1^{-}(S_k, \Sigma_k) , in order to illustrate a general method which should apply also to non variational perturbations of S_k , we will give an alternative proof of the existence of \psi_1 by maximum principle methods for \Sigma_k -admissible viscosity solutions.
Since the complete argument to solve (1.8) is somewhat involved, perhaps it is worth giving the general idea first. We will show that \psi_1 \in C(\overline{\Omega}) is the limit as n \to +\infty (up to an extracted subsequence) of the normalized solutions
w_n : = \frac{v_n}{||v_n||_{\infty}} |
where each v_n \in C(\overline{\Omega}) is a \Sigma_k -admissible viscosity solution of the auxiliary problem
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 v_n) = 1 - \lambda_n v_{n} |v_{n}|^{k-1} & {\rm in} \ \Omega \\ v_n = 0 & {\rm on} \ \partial \Omega \end{array} \right. \end{equation} | (6.1) |
and \{ \lambda_n \}_{n \in {\mathbb N}} is any fixed sequence of spectral parameters with 0 < \lambda_n \nearrow \lambda_1^{-} as n \to + \infty . The existence of the solutions v_n to (6.1) presents the same difficulties as mentioned above for (1.8), but for each fixed \lambda \in (0, \lambda_1^-) we will show that the problem
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 u) = 1 - \lambda u |u|^{k-1} & {\rm in} \ \Omega \\ u = 0 & {\rm on} \ \partial \Omega \end{array} \right. \end{equation} | (6.2) |
has a \Sigma_k -admissible solution u \in C(\overline{\Omega}) by an inductive procedure starting from u_0 = 0 and then solving
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 u_n) = 1 - \lambda u_{n-1} |u_{n-1}|^{k-1} : = f_n & {\rm in} \ \Omega \\ u_n = 0 & {\rm on} \ \partial \Omega \end{array} \right. \end{equation} | (6.3) |
for a decreasing sequence of \Sigma_k -admissible solutions \{u_n\}_{n \in {\mathbb N}} \subset C(\overline{\Omega}) (which are negative in \Omega ) and then pass to the limit as n \to +\infty . Notice that the equation in (6.3) is proper as u_n does not appear explicitly and hence the equation is non increasing in u_n . Moreover, it will turn out that one can pass to the limit along a subsequence provided that there is a uniform Hölder bound on ||u_n||_{C^{0, \alpha}(\overline{\Omega})} for each n \in {\mathbb N} and some \alpha \in (0, 1] .
We begin with the following existence and uniqueness result for the underlying degenerate elliptic Dirichlet problems in (6.3) in the nonlinear case k \in \{ 2, \ldots, N\} . While this result is not new, for completeness we prefer to discuss it.
Theorem 6.1. Let \Omega be a strictly (k-1) -convex domain of class C^2 and let f \in C(\overline{\Omega}) be a nonnegative function. There exists a unique k -convex solution u \in C(\overline{\Omega}) of the Dirichlet problem
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 u) = f & {\rm in} \ \Omega \\ u = 0 & {\rm on} \ \partial \Omega. \end{array} \right. \end{equation} | (6.4) |
More precisely, there is \Sigma_k -admissible solution u \in C(\overline{\Omega}) of S_k(D^2 u) - f(x) = 0 in \Omega in the sense of Definition 2.5(c) such that u = 0 on \partial \Omega .
Proof. The existence and uniqueness for \Sigma_k -admissible viscosity solutions follows from the main results in [6]. See Theorem 1.2 as applied in section 5 of that paper. When f > 0 , one has smooth solutions if \partial \Omega is smooth as follows from [5].
Briefly, we give an idea of the proof for completeness sake. A \Sigma_k -admissible viscosity solution of (6.4) is a \Theta_k -harmonic function which vanishes on the boundary where \Theta_k: \Omega \to \mathcal{S}(N) is the uniformly continuous elliptic map defined by
\begin{equation} \Theta_k(x): = \{ A \in \Sigma_k: \ F_k(x,A): = S_k(A) - f(x) \geq 0\} \ \ \text{for each} \ x \in \Omega. \end{equation} | (6.5) |
The uniform continuity is with respect to the Hausdorff distance on \mathcal{S}(N) and follows from the uniform continuity of f \in C(\overline{\Omega}) . Using Propositions 5.1 and 5.3 of [6], one has the equivalence between u \in C(\Omega) being \Theta_k -harmonic and u being a \Sigma_k -admissible viscosity solution of F_k(x, D^2u) = 0 since one can easily verify the needed structural conditions ((1.14)–(1.16) and (1.18)); that is,
\begin{equation} F_k(x,A + P) \geq F_k(x,A) \ \ \text{for each} \ x \in \Omega, A \in \Sigma_k, P \in {\mathcal P}; \end{equation} | (6.6) |
\begin{equation} \text{for each} \ x \in \Omega \ \text{there exists} \ A \in \Sigma_k \ \text{such that} \ F_k(x,A) = 0; \end{equation} | (6.7) |
\begin{equation} \partial \Sigma_k \subset \{A \in \mathcal{S}(N): \ F_k(x,a) \leq 0\} \ \text{for each} \ x \in \Omega; \end{equation} | (6.8) |
and
\begin{equation} F_k(x,A) \gt 0 \ \ \text{for each} \ x \in \Omega \ \text{and each} \ A \in \Theta_k(x)^{\circ}. \end{equation} | (6.9) |
Conditions (6.6)–(6.8) say that \Theta_k defined by (6.5) is an elliptic branch of the equation F_k(x, D^2u) = 0 in the sense of Kyrlov [23] (see Proposition 5.1 of [6]) and the non-degeneracy condition (6.9) ensures that \Theta_k -superharmonics are \Sigma_k -admissible viscosity supersolutions of F_k(x, D^2u) = 0 (see Proposition 5.3 of [6]). Finally the existence of a unique u \in C(\overline{\Omega}) which is \Theta_k -harmonic taking on the continuous boundary value \varphi \equiv 0 follows from Perron's method (Theorem 1.2 of [6]) since \Theta_k is uniformly continuous and the strict (k-1) -convexity implies the needed strict \overrightarrow{{\Sigma}}_k and \overrightarrow{{\widetilde{\Sigma}}}_k convexity (which is the content of Proposition 2.11).
Remark 6.2. Using the language of Harvey-Lawson [16], one could also say that (S_k, \Sigma_k) is a compatible operator-subequation pair (see Definition 2.4 of [16]) and since the continuous boundary data \varphi \equiv 0 has its values in S_k(\Sigma_k) , the result follows also from Theorem 2.7 of [16].
Next we discuss the global Hölder regularity of the unique solution to Theorem 6.1 in the case k > N/2 , which will lead to compactness for bounded sequences of solution.
Theorem 6.3. Under the assumptions of Theorem 6.1, if k > N/2 then the unique solution u to the Dirichlet problem (6.4) belongs to C^{0, \alpha}(\overline{\Omega}) with \alpha : = 2 - N/k > 0 . In particular, there exists C > 0 which depends on \Omega, \alpha and \sup_{\Omega}(-u) such that
\begin{equation} |u(x) - u(y)| \leq C \, |x - y|^{\alpha}, \ \ \forall \ x, y \in \overline{\Omega}. \end{equation} | (6.10) |
Before giving the proof, we formalize a few observations concerning the restriction k > N/2 in the statement.
Remark 6.4. For the proof of the global Hölder bound (6.10), we will adapt the technique developed in the celebrated paper of Ishii and Lions [17]. The key step involves a uniform local interior estimate which uses a comparison principle argument for the solution u (which is \Sigma_k -subharmonic since f \geq 0 ) and a family of comparison functions defined in terms of the auxiliary function \phi(x): = |x|^\alpha , where \alpha \in (0, 1] . One needs that \phi is \Sigma_k -superharmonic on its domain. It is known that for \alpha = 2-\frac{N}{k} , the function \phi is a classical \Sigma_k -harmonic away from the origin, but \alpha > 0 requires the condition k > N/2 . This restriction can be interpreted in terms of the Riesz characteristic of the closed convex cone \Sigma_k \subset \mathcal{S}(N) as described in Harvey-Lawson [15]. Using the measure theoretic techniques developed by Trudinger and Wang [29,30] and Labutin [24], perhaps it is possible to obtain the global Hölder bound (6.10) if k \leq N/2 . However, our intended focus is limited to maximum principle techniques and hence we have not pursue such improvements here.
Proof of Theorem 6.3. Since u \in C(\overline{\Omega}) by Theorem 6.1, the claim that u \in C^{0, \alpha}(\overline{\Omega}) , reduces to proving the estimate (6.10). Notice that u is k -convex (it is a \Sigma_k -admissible subsolution) and u vanishes on the boundary and hence u \leq 0 in \Omega by Theorem 3.2. If f \equiv 0 , then u \equiv 0 and the conclusion of Theorem 6.3 holds trivially. Otherwise, u < 0 in \Omega by Theorem 3.2 and
\begin{equation} ||u||_{\infty} : = \sup\limits_{\Omega}|u| = \sup\limits_{\Omega}(-u). \end{equation} | (6.11) |
For the Hölder estimate (6.10), it suffices to find \rho > 0 and C_{\rho} > 0 for which
\begin{equation} |u(x) - u(y)| \leq C_{\rho} \, |x - y|^{\alpha}, \ \ \forall \ x, y \in \overline{\Omega} \ \ \text{with} \ |x - y| \lt \rho. \end{equation} | (6.12) |
In fact, as is well known, if (6.12) holds, then using the boundedness of u one has
\sup\limits_{\substack{x, y \in \overline{\Omega} \\ x \neq y}} \frac{|u(x) - u(y)|}{|x-y|^{\alpha}} \leq \max \left\{ \frac{2 ||u||_{\infty}}{\rho^{\alpha}}, C_{\rho} \right\}. |
In order to prove (6.12), first consider the case when y lies on \partial \Omega (the argument for x \in \partial \Omega is the same). In this case, u(y) = 0 and u(x) \leq 0 . Then the boundary estimate of Proposition 4.3 shows that y \in \partial \Omega and each x \in \Omega_{d_0} one has
\begin{equation} |u(y) - u(x)| = -u(x) \leq C_3 \, d(x) = C_3 \, \min\limits_{z \in \partial \Omega} |x-z| \leq C_3 |x-y|. \end{equation} | (6.13) |
By choosing
\begin{equation} \rho \leq \min \{ d_0, 1 \} \ \ \text{and} \ \ C_{\rho} \geq C_3 \end{equation} | (6.14) |
one has (6.12) for each \alpha \in (0, 1] if y (or x ) lies on the boundary.
Next, let y \in \Omega and consider the comparison function
\begin{equation} v_y(x): = u(y) + C_{\rho} \, |x-y|^{\alpha} \end{equation} | (6.15) |
One wants determine \rho > 0 sufficiently small and C_{\rho} > 0 sufficiently large (recall the restrictions (6.14)) so that
\begin{equation} u(x) \leq v_y(x) \ \ \text{for each} \ x \in \Omega \cap B_{\rho}(y) \end{equation} | (6.16) |
and hence
\begin{equation} u(x) - u(y) \leq C_{\rho} \, |x-y|^{\alpha} \ \ \text{for each} \ x \in \Omega \cap B_{\rho}(y). \end{equation} | (6.17) |
Then, by exchanging the roles of x and y , one would have
\begin{equation} |u(x) - u(y)| \leq C_{\rho} \, |x-y|^{\alpha} \ \ \text{for each} \ x,y \in \Omega \ \text{with} \ |x-y| \lt \rho, \end{equation} | (6.18) |
which would then complete the proof.
In order to establish (6.16), notice that u is a \Sigma_k -admissible solution of S_k(D^2u) = f \geq 0 in \Omega . In particular, u is \Sigma_k -subharmonic ( k -convex) in \Omega . Moreover, for k > N/2 one knows that for each y \in {\mathbb R}^N , the function defined by
\begin{equation} w_k(x): = |x -y|^{2-N/k} \ \ \text{for} \ x \in {\mathbb R}^N \setminus \{y\} \end{equation} | (6.19) |
is smooth, k -convex and satisfies S_k(w_k) \equiv 0 on its domain (see section 2 of [30]). The same is obviously true for the translated version v_y of (6.15) with the choice \alpha: = 2 - N/k when k > N/2 . Indeed, using the radial formula (2.70) of Lemma 2.14 with h(r): = u(y) + C_{\rho} \, r^{\alpha} one finds that
\begin{equation} S_j(D^2v_y(x)) = \left( C_{\rho} \, \alpha |x - y|^{\alpha - 2} \right)^j \frac{(N -1)!}{j! \, (N-j)!} \left[ (\alpha - 2)j + N \right] \ \ \text{for each} \ x \neq y. \end{equation} | (6.20) |
When \alpha = 2 - N/k , this is positive for every j = 1, \ldots, k-1 and it vanishes for j = k . In particular, v_y is \Sigma_k -superharmonic in every punctured ball \dot{B}_{\rho}(y) = B_{\rho}(y) \setminus \{y\} . Hence, by the comparison principle (Theorem 3.1) for \Sigma_k sub and superharmonics we will have
\begin{equation} u(x) \leq v_y(x) \ \ \text{for each} \ x \in \Omega \cap \dot{B}_{\rho}(y) \end{equation} | (6.21) |
provided that
\begin{equation} u \leq v_y \ \ \text{on} \ \partial(\Omega \cap \dot{B}_{\rho}(y)) \end{equation} | (6.22) |
where \partial(\Omega \cap \dot{B}_{\rho}(y)) = \{y\} \cup (\partial B_{\rho}(y) \cap \Omega) \cup (\partial \Omega \cap \overline{B}_{\rho}(y)) .
We analyze the three possibilities. At the point y , one has
u(y) = v_y(y) = u(y) + C_{\rho} \, |y - y|. |
Next, for x \in \partial \Omega \cap \overline{B}_{\rho}(y) (which is empty if B_{\rho}(y) \subset \subset \Omega ) one has
u(x) = 0 \ \ \text{while} \ v_y(x) = u(y) + C_{\rho} \, |x-y|^{\alpha}, |
where u(y) < 0 for y \in \Omega as noted above. Since x \in \partial \Omega , with \rho \leq d_0 the condition |x-y| < \rho means that y \in \Omega_{d_0} and one can again use the boundary estimate of Proposition 4.3 to estimate u(y) from below
v_y(x) \geq -C_3 \, |x-y| + C_{\rho} \, |x-y|^{\alpha} \geq (C_{\rho} - C_3)|x - y|^{\alpha} \geq 0, |
provided that \rho \leq 1 and C_{\rho} \geq C_3 as in (6.14). Finally, if x \in \partial B_{\rho}(y) \cap \Omega we will have v_y(x) = u(y) + C_{\rho} \, |x-y|^{\alpha} \geq u(x) if
|u(x) - u(y)| \leq C_{\rho} \, \rho^{\alpha}. |
Having now fixed \rho \leq \min \{d_0, 1\} , since |u(x) - u(y)| \leq 2 ||u||_{\infty} it is enough to choose
\begin{equation} C_{\rho} \geq \frac{2 ||u||_{\infty}}{\rho^{\alpha}}, \end{equation} | (6.23) |
in addition to C_{\rho} \geq C_3 .
We now implement the iteration scheme (sketched above) to prove the existence of a principal eigenfunction \psi_1 associated to \lambda_1^{-} = \lambda_1^{-}(S_k, \Sigma_k) in the "regular case" with k > N/2 .
Remark 6.5. We will make use of the fact that the sets of \Sigma_k -subharmonic and \Sigma_k -superharmonic functions on \Omega are closed under the operation of taking uniform limits in \Omega of sequences. See property (5)' in [11], for example.
Theorem 6.6. Suppose that k > N/2 . Let \Omega be a strictly (k-1) -convex domain of class C^2 . If \{v_n\}_{n \in {\mathbb N}} is the sequence of k -convex solutions of (6.1) with 0 < \lambda_n \nearrow \lambda_1^{-} as n \to + \infty , then the normalized sequence defined by w_n: = v_n/||v_n||_{\infty} admits a subsequence which converges uniformly to a principal eigenfunction \psi_1 for (1.8), which is negative on \Omega .
Proof. We divide the proof into two big steps, with several claims to be justified.
Step 1: For each \lambda \in (0, \lambda_1^-) , show that there exists a \Sigma_k -admissible solution u to the Dirichlet problem (6.24); that is,
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 u) = 1 - \lambda u |u|^{k-1} & {\rm in} \ \Omega \\ u = 0 & {\rm on} \ \partial \Omega. \end{array} \right. \end{equation} | (6.24) |
As indicated above, we will look for u as a decreasing limit of solutions \{u_n\}_{n \in {\mathbb N}_0} of the Dirichlet problem (6.3), that is,
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 u_n) = 1 - \lambda u_{n-1} |u_{n-1}|^{k-1} : = f_n & {\rm in} \ \Omega \\ u_n = 0 & {\rm on} \ \partial \Omega \end{array} \right. \end{equation} | (6.25) |
With u_0 \equiv 0 , we apply Theorem 6.1 to find u_1 \in C(\overline{\Omega}) a \Sigma_k -admissible solution of
\mbox{ $S_k(D^2u_1) = 1$ in $\Omega$ and $u_1 = 0$ on $\partial \Omega$.} |
Since u_1 is a \Sigma_k -admissible solution it is necessarily k -convex and hence satisfies the (strong) maximum principle so that u_1 \leq 0 on \overline{\Omega} (and u_1 < 0 in \Omega ) and hence
f_2: = 1 - \lambda u_1 |u_1|^{k-1} = 1 + \lambda |u_1|^k \geq 0 |
and the induction proceeds using Theorem 6.1 to produce the sequence \{u_n\}_{n \in {\mathbb N}_0} of non-positive \Sigma_k -admissible solutions which also satisfy
\begin{equation} u_n \lt 0 \ \ \text{on} \ \Omega \ \text{for each} \ n \in {\mathbb N}. \end{equation} | (6.26) |
Claim 1: \{u_n\}_{n \in {\mathbb N}_0} is a decreasing sequence of k -convex functions.
By construction, all of the functions vanish on \partial \Omega and are negative in \Omega for n \geq 1 . We use induction. As we have seen u_1 < 0 : = u_0 on \Omega . Assuming that u_{n} \leq u_{n-1} on \Omega , we need to show that u_{n+1} \leq u_n on \Omega . We have that u_{n+1} is a \Sigma_k -admissible solution of
S_k(D^2 u_{n+1}) = 1 - \lambda u_n |u_n|^{k-1} = 1 + \lambda |u_n|^{k} \ \ \text{in} \ \Omega, |
where we have again used u_n \leq 0 , but then the inductive hypothesis yields
\begin{equation} u_{n} \leq u_{n-1} \leq 0 \ \ \Rightarrow \ \ |u_n| = -u_n \geq -u_{n-1} = |u_{n-1}| \end{equation} | (6.27) |
and hence
S_k(D^2 u_{n+1}) \geq 1 + \lambda |u_{n-1}|^k = f_n = S_k(D^2u_n). |
By the comparison principle, one concludes that u_{n+1} \leq u_n on \Omega .
Claim 2: The sequence \{u_n\}_{n \in {\mathbb N}} is bounded in sup norm; that is, there exists M > 0 such that
\begin{equation} ||u_n||_{\infty} = \sup\limits_{\Omega}(-u_n) \leq M \ \ for \;each \ n \in {\mathbb N}. \end{equation} | (6.28) |
We argue by contradiction, assuming that the increasing sequence ||u_n||_{\infty} satisfies \lim_{n \to +\infty} ||u_n||_{\infty} = +\infty . Since u_n < 0 on \Omega for each n \in {\mathbb N} , we can define
\begin{equation} v_n: = \frac{u_n}{||u_n||_{\infty}} \ \ \text{so that} \ \ ||v_n||_{\infty} = 1 \ \ \text{for each} \ n \in {\mathbb N}. \end{equation} | (6.29) |
Since the equation S_k(D^2u_n) = 1 + \lambda |u_{n-1}|^k is homogeneous of degree k , one has
\begin{equation} S_k(D^2v_n) = \frac{1}{||u_n||_{\infty}^k} + \lambda \frac{ ||u_{n-1}||_{\infty}^k}{||u_n||_{\infty}^k} \, |v_{n-1}|^k. \end{equation} | (6.30) |
Now, making again use of the negativity and monotonicity in (6.27) one has
\begin{equation} \beta_n: = \frac{||u_{n-1}||_{\infty}^k}{||u_n||_{\infty}^k} \in (0, 1] \end{equation} | (6.31) |
and combining (6.30) with (6.31) yields
\begin{equation} S_k(D^2 v_n) = \frac{1}{||u_n||_{\infty}^k} + \lambda \beta_n |v_{n-1}|^k : = g_n, \end{equation} | (6.32) |
where g_n\in C(\overline{\Omega}) and is non-negative. Since k > N/2 and since the global Hölder bound of Theorem 6.3 depends only on \Omega, \alpha and ||v_n||_{\infty} \equiv 1 , the sequence of solutions \{v_n\}_{n \in {\mathbb N}} is bounded in C^{0, 2 - N/k}(\overline{\Omega}) and hence admits v \in C(\overline{\Omega}) and a subsequence such that
\begin{equation} v_{n_j} \to v \ \ \text{uniformly on} \ \overline{\Omega}. \end{equation} | (6.33) |
In addition, 0 < \beta_{n_j} \leq 1 is increasing so converges to some \beta_{\infty} \in (0, 1] . The uniform limit v is a \Sigma_k -admissible (super)solution of
\mbox{ $S_k(D^2 v) + \lambda \beta_{\infty} v |v|^{k-1} = 0$ in $\Omega$ $v = 0$ on $\partial \Omega$,} |
where \lambda \beta_{\infty} \leq \lambda < \lambda_1^- . By the minimum principle characterization of \lambda_1^- , we must have v \geq 0 in \Omega . However, each u_n \in C(\overline{\Omega}) is negative in \Omega and hence has a negative minimum at some interior point x_n \in \Omega and hence v_n(x_n) = -1 for each n which contradicts the fact that uniform limit of (6.33) satisfies v \geq 0 on \Omega .
Claim 3: The sequence \{ u_n \}_{n \in {\mathbb N}} admits a subsequence \{u_{n_j}\}_{j \in {\mathbb N}} \subset C(\overline{\Omega}) which converges uniformly on \overline{\Omega} to a \Sigma_k -admissible solution u of (6.24).
Exploiting the boundedness of Claim 3 for the sequence \{ u_n \}_{n \in {\mathbb N}} , we can use the same argument involving the global Hölder estimate of Theorem 6.3 to extract a uniformly convergent subsequence with limit u \in C(\overline{\Omega}) with limit u which is a \Sigma_k -admissible solution of (6.24). This completes Step 1 of the proof.
Step 2: Show that there exists \psi_1 \in C(\overline{\Omega}) which is negative in \Omega and is a \Sigma_k -admissible solution of (1.8); that is,
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 \psi_1) + \lambda_1^{-}\psi_1 |\psi_1|^{k-1} = 0 & {\rm in} \ \Omega \\ \psi_1 = 0 & {\rm on} \ \partial \Omega. \end{array} \right. \end{equation} | (6.34) |
Consider a sequence \{ \lambda_n\}_{n \in {\mathbb N}} \in (0, \lambda_1^-) with \lambda_n \nearrow \lambda_1^- and the associated sequence \{ v_n\}_{n \in {\mathbb N}} \subset C(\overline{\Omega}) of solutions to (6.24) with \lambda = \lambda_n ; that is,
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 v_n) = 1 - \lambda_n v_n |v_n|^{k-1} & {\rm in} \ \Omega \\ v_n = 0 & {\rm on} \ \partial \Omega. \end{array} \right. \end{equation} | (6.35) |
Since each v_n is \Sigma_k -subharmonic in \Omega and vanishes on the boundary, v_n < 0 on \Omega for each n .
Claim 4: One has ||v_n||_{\infty} \to + \infty as n \to +\infty .
We argue by contradiction. If not, then again by the global Hölder bound of Theorem 6.3 we can extract a subsequence of these functions which are \Sigma_k -subharmonic and negative in \Omega and which converges uniformly on \overline{\Omega} to a \Sigma_k -admissible solution w \in C(\overline{\Omega}) to the Dirichlet problem
\begin{equation} \left\{ \begin{array}{cc} S_k(D^2 w) + \lambda_1^- w |w|^{k-1} = 1 & {\rm in} \ \Omega \\ w = 0 & {\rm on} \ \partial \Omega. \end{array} \right. \end{equation} | (6.36) |
Since w \in C(\overline{\Omega}) is non-positive, there exists \varepsilon > 0 such that
\begin{equation} - \varepsilon w |w|^{k-1} \leq 1 \ \ \text{in} \ \overline{\Omega}. \end{equation} | (6.37) |
Hence w is a k -convex, negative in \Omega and satisfies (in the \Sigma_k -admissible viscosity sense)
\begin{equation} S_k(D^2 w) + (\lambda_1^- + \varepsilon) w |w|^{k-1} \geq 0, \end{equation} | (6.38) |
which contradicts the Definition 5.1 of \lambda_1^-(S_k, \Sigma_k) which is finite by Theorem 7.1. Hence Claim 4 holds.
Finally, consider the normalized sequence defined by w_n: = v_n/||v_n||_{\infty} which are \Sigma_k -admissible viscosity solutions of
\begin{equation} S_k(D^2 w_n) + \lambda_n w_n |w_n|^{k-1} = \frac{1}{||v_n||_{\infty}} \ \ \text{in} \ \Omega \ \ \ \ \text{and} \ w_n = 0 \ \ \text{on} \ \partial \Omega. \end{equation} | (6.39) |
The uniformly bounded sequence \{w_n\}_{n \in {\mathbb N}} \subset C(\overline{\Omega}) admits a subsequence which converges uniformly on \overline{\Omega} some \psi_1 \in C(\overline{\Omega}) which is a \Sigma_k -admissible solution of the eigenvalue problem (1.8) as \frac{1}{||v_n||_{\infty}} \to 0 as n \to + \infty .
In this section, we will provide an upper bound for the generalized principle eigenvalue \lambda_1^-(S_k, \Sigma_k) as defined in Definition 5.1. Recall that the lower bound
\begin{equation} \lambda_1^-(S_k, \Sigma_k) \geq 2^k C_{N,k}R^{-2k} \ \ \text{with} \ \ C_{N,k} = \left( \begin{array}{c} N \\ k \end{array} \right) \end{equation} | (7.1) |
was given in Lemma 5.2 for bounded domains \Omega which are contained in a ball B_R(0) .
An upper bound will be found by constructing a suitable test function which contradicts the minimum principle of Theorem 5.4 on a ball B_R(0) \subset \Omega and makes use of the monotonicity of \lambda_1^-(S_k, \Sigma_k) with respect to set inclusion. More precisely, if we denote by \lambda_1^-(\Omega) the generalized principal eigenvalue \lambda_1^-(S_k, \Sigma_k) with respect to the bounded domain \Omega then one has that
\begin{equation} \Omega^{\prime} \subset \Omega \ \ \Rightarrow \ \ \lambda_1^- (\Omega) \leq \lambda_1^- (\Omega^{\prime}). \end{equation} | (7.2) |
Indeed, since
\lambda_1^{-}(\Omega): = \sup \{ \lambda \in {\mathbb R}: \ \exists \, \psi \in \Phi_k^{-}(\Omega) \ \text{with} \ S_k(D^2 \psi) + \lambda \psi |\psi|^{k-1} \geq 0 \ \text{in} \ \Omega\}, |
if \lambda admits such a \psi for \Omega then it also admits \psi for \Omega^{\prime} and hence (7.2) holds. Our upper bound is contained in the following theorem.
Theorem 7.1. If a bounded domain \Omega contains the ball B_R = B_R(0) , then
\begin{equation} \lambda_1^{-}(\Omega) \leq 2^k \gamma_{N,k} C_{N,k} R^{-2k} \end{equation} | (7.3) |
where
\begin{equation} C_{N,k} = \left( \begin{array}{c} N \\ k \end{array} \right) \quad and \quad \gamma_{N,k} : = \frac{1}{ N} \left( \frac{N + 2k}{k + 1} \right)^{k+1}. \end{equation} | (7.4) |
Proof. Consider the radial test function (as used in [5]) defined by
\begin{equation} u(x) : = - \frac{1}{4} \left(R^2 - |x|^2 \right)^2 \end{equation} | (7.5) |
and let r: = |x| . It suffices to show that u is a \Sigma_k -admissible supersolution of
\begin{equation} S_k(D^2 u) + \lambda u |u|^{k-1} = 0 \ \text{in} \ B_R \end{equation} | (7.6) |
with
\begin{equation} \lambda = 2^k \gamma_{N,k} C_{N,k} R^{-2k}. \end{equation} | (7.7) |
Indeed, notice that:
\begin{equation} u \in C^{\infty}( {\mathbb R}^N) \ \ \text{and hence} \ u \in \mathrm{LSC}(\overline{B}_R); \end{equation} | (7.8) |
\begin{equation} u = 0 \ \text{on} \ \partial {B}_R; \end{equation} | (7.9) |
and
\begin{equation} \mbox{$\partial B_R \in C^{\infty}$ and $B_R$ is strictly $(k-1)$-convex for each $k \in \{1, \ldots , N \}$.} \end{equation} | (7.10) |
However,
\begin{equation} u \lt 0 \ \text{on} \ B_R \end{equation} | (7.11) |
and hence u does not satisfy the minimum principle of of Theorem 5.4 on B_R and hence one must have
\lambda = 2^k \gamma_{N,k} C_{N,k} R^{-2k} \geq \lambda_1^{-}(B_R) \geq \lambda_1^{-}(\Omega), |
which would complete the proof.
Since u \in C^2(B_R) , it will be a \Sigma_k -admissible supersolution of (7.6) provided that
\begin{equation} S_k(D^2 u(x)) + \lambda u(x) |u(x)|^{k-1} \leq 0 \ \text{for each} \ x \in B_R, \end{equation} | (7.12) |
as follows easily from Remark 2.7 (b) by taking \varphi = u as the lower test function. Using the radial formula (2.71) with h(r): = -(R^2-r^2)^2/4 one computes to find
\begin{equation} S_k(D^2u) + \lambda u |u|^{k-1} = \left( \begin{array}{c} N \\ k \end{array} \right) (R^2 - r^2)^{k-1} g(r) \end{equation} | (7.13) |
where
\begin{equation} g(r) = R^2 - \left( 1 + \frac{2k}{N} \right) r^2- \frac{\gamma_{N,k}}{2^k R^{2k}} (R^2 - r^2)^{k+1}. \end{equation} | (7.14) |
We will have the needed inequality (7.12) if we show that g defined in (7.14) satisfies
\begin{equation} g(r) \leq 0 \ \ \text{for each} \ r \in [0,R) \ \ \text{where again} \ \gamma_{N,k} = \frac{1}{N} \left( \frac{N + 2k}{k + 1} \right)^{k+1} . \end{equation} | (7.15) |
Notice that g(R) = -\frac{2k}{N} R^2 < 0 and g(0) = R^2(1 - 2^{-k} \gamma_{N, k}) \leq 0 provided that
\begin{equation} \widetilde{\gamma}_{N,k} : = 2^{-k} \gamma_{N,k} = \frac{1}{2^k N} \left( \frac{N + 2k}{k + 1} \right)^{k+1} \geq 1 \ \ \text{for each} \ k \in \{ 1, \ldots, N\}. \end{equation} | (7.16) |
The lower bounds in (7.16) do hold. To see this, the cases N = 1 and N = 2 are easily checked. Next, a simple computation shows that \widetilde{\gamma}_{N, 1} \geq 1 for all N . Finally, rewriting \widetilde{\gamma}_{N, k} as
\widetilde{\gamma}_{N,k} = \frac{2}{N} \left( 1 + \frac{N/2 - 1}{k+1} \right)^{k+1}, |
one sees that \widetilde{\gamma}_{N, k} strictly increasing in k if N \geq 3 and hence \widetilde{\gamma}_{N, k} \geq \widetilde{\gamma}_{N, 1} \geq 1 also for N \geq 3 .
It remains to check that g \leq 0 on (0, R) and we simplify notation by setting \gamma: = \gamma_{N, k} . Since
\begin{equation} g'(r) = -2r \left[ 1 + \frac{2k}{N} - \frac{\gamma(k+1)}{2^k R^{2k}} (R^2 - r^2)^{k} \right] \end{equation} | (7.17) |
elementary calculus shows that there is a unique \bar{r} \in (0, R) such that
\begin{equation} g'(\bar{r}) = 0, \ \ g'(r) \gt 0 \ \text{for} \ r \in (0, \bar{r}) \ \ \text{and} \ g'(r) \lt 0 \ \text{for} \ r \in (\bar{r}, R). \end{equation} | (7.18) |
Hence we just need to show that g(\bar{r}) \leq 0 . From (7.17), the critical value \bar{r} in (7.18) satisfies the relations
\begin{equation*} \frac{\gamma}{2^k R^{2k}}(R^2 -\bar{r}^2)^k = \frac{N + 2k}{N(k+1)} \quad \text{and} \quad \bar{r}^2 = (1 - \delta)R^2 \ \ \text{with} \ \delta: = 2 \left[ \frac{N + 2k}{N(k+1)} \frac{1}{\gamma} \right]^{1/k}, \end{equation*} |
and hence
\begin{equation*} g(\bar{r}) = R^2 \left( - \frac{2k}{N} + \delta \frac{(N + 2k)k}{N(k+1)} \right) \leq 0 \end{equation*} |
provided that
\delta = 2 \left[ \frac{N + 2k}{N(k+1)} \frac{1}{\gamma} \right]^{1/k} \leq \frac{2(k+1)}{N + 2k} \ \ \Longleftrightarrow \ \ \gamma \geq \gamma_{N,k} = \frac{1}{N} \left( \frac{N + 2k}{k + 1} \right)^{k+1} . |
The authors wish to thank an anonymous referee for the careful reading and suggestions which led to improvement of the original manuscript.
Payne is partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and the projects: GNAMPA 2017 "Viscosity solution methods for fully nonlinear degenerate elliptic equations", GNAMPA 2018 "Costanti critiche e problemi asintotici per equazioni completamente non lineari" e GNAMPA 2019 "Problemi differenziali per operatori fully nonlinear fortemente degeneri".
The authors declare no conflict of interest.
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