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Dedicated to Giuseppe Mingione on the occasion of his 50th birthday, with admiration.
In this paper we provide a limiting partial regularity criterion for vector-valued minimizers u:Ω⊂Rn→RN, n≥2, N>1, of nonhomogeneous, quasiconvex variational integrals as:
W1,p(Ω;RN)∋w↦F(w;Ω):=∫Ω[F(Dw)−f⋅w]dx, | (1.1) |
with standard p-growth. More precisely, we infer the optimal [31, Section 9] ε-regularity condition
supBϱ⋐Ωϱm−∫Bϱ|f|mdx≲ε⟹ Du has a.e. bounded mean oscillation, |
and the related borderline function space criterion
f∈L(n,∞) ⟹ supBϱ⋐Ωϱm−∫Bϱ|f|mdx≲ε. |
This is the content of our main theorem.
Theorem 1.1. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let u∈W1,p(Ω,RN) be a local minimizer of functional (1.1). Then, there exists a number ε∗≡ε∗(data)>0 such that if
‖f‖Ln,∞(Ω)≤(|B1|4n/m)1/nε∗, | (1.2) |
then there exists an open set Ωu⊂Ω with |Ω∖Ωu|=0 such that
Du∈BMOloc(Ωu;RN×n). | (1.3) |
Moreover, the set Ωu can be characterized as follows
Ωu:={x0∈Ω:∃εx0,ϱx0>0such that E(u;Bϱ(x0))≤εx0 for some ϱ≤ϱx0}, |
where E(⋅) is the usual excess functional defined as
E(w,z0;Bϱ(x0)):=( −∫Bϱ(x0)|z0|p−2|Dw−z0|2+|Dw−z0|p dx)1p. | (1.4) |
We immediately refer to Section 1.2 below for a description of the structural assumptions in force in Theorem 1.1. Let us put our result in the context of the available literature. The notion of quasiconvexity was introduced by Morrey [38] in relation to the delicate issue of semicontinuity of multiple integrals in Sobolev spaces: an integrand F(⋅) is a quasiconvex whenever
−∫B1(0)F(z+Dφ)dx≥F(z)holds for all z∈RN×n, φ∈C∞c(B1(0),RN). | (1.5) |
Under power growth conditions, (1.5) is proven to be necessary and sufficient for the sequential weak lower semicontinuity on W1,p(Ω;RN); see [1,4,35,36,38]. It is worth stressing that quasiconvexity is a strict generalization of convexity: the two concepts coincide in the scalar setting (N=1), or for 1-d problems (n=1), but sharply differ in the multidimensional case: every convex function is quasiconvex thanks to Jensen's inequality, while the determinant is quasiconvex (actually polyconvex), but not convex, cf. [24, Section 5.1]. Another distinctive trait is the nonlocal nature of quasiconvexity: Morrey [38] conjectured that there is no condition involving only F(⋅) and a finite number of its derivatives that is both necessary and sufficient for quasiconvexity, fact later on confirmed by Kristensen [29]. A peculiarity of quasiconvex functionals is that minima and critical points (i.e., solutions to the associated Euler-Lagrange system) might have very different behavior under the (partial) regularity viewpoint. In fact, a classical result of Evans [22] states that the gradient of minima is locally Hölder continuous outside a negligible, " singular" set, while a celebrated counterexample due to Müller and Šverák [39] shows that the gradient of critical points may be everywhere discontinuous. After Evans seminal contribution [22], the partial regularity theory was extended by Acerbi and Fusco [2] to possibly degenerate quasiconvex functionals with superquadratic growth, and by Carozza, Fusco and Mingione [8] to subquadratic, nonsingular variational integrals. A unified approach that allows simultaneously handling degenerate/nondegenerate, and singular/nonsingular problems, based on the combination of A-harmonic approximation [21], and p-harmonic approximation [20], was eventually proposed by Duzaar and Mingione [19]. Moreover, Kristensen and Mingione [30] proved that the Hausdorff dimension of the singular set of Lipschitz continuous minimizers of quasiconvex multiple integrals is strictly less than the ambient space dimension n, see also [5] for further developments in this direction. We refer to [3,15,16,25,26,27,28,37,41,42] for an (incomplete) account of classical, and more recent advances in the field. In all the aforementioned papers are considered homogeneous functionals, i.e., f≡0 in (1.1). The first sharp ε-regularity criteria for nonhomogeneous quasiconvex variational integrals guaranteeing almost everywhere gradient continuity under optimal assumptions on f were obtained by De Filippis [12], and De Filippis and Stroffolini [14], by connecting the classical partial regularity theory for quasiconvex functionals with nonlinear potential theory for degenerate/singular elliptic equations, first applied in the context of partial regularity for strongly elliptic systems by Kuusi and Mingione [33]. Potential theory for nonlinear PDE originates from the classical problem of determining the best condition on f implying gradient continuity in the Poisson equation −Δu=f, that turns out to be formulated in terms of the uniform decay to zero of the Riesz potential, in turn implied by the membership of f to the Lorentz space L(n,1), [9,31]. In this respect, a breakthrough result due to Kuusi and Mingione [32,34] states that the same is true for the nonhomogeous, degenerate p-Laplace equation–in other words, the regularity theory for the nonhomogeneous p-Laplace PDE coincides with that of the Poisson equation up to the C1-level. This important result also holds in the case of singular equations [18,40], for general, uniformly elliptic equations [6], up to the boundary [10,11,13], and at the level of partial regularity for p-Laplacian type systems without Uhlenbeck structure, [7,33]. We conclude by highlighting that our Theorem 1.1 fits this line of research as, it determines for the first time in the literature optimal conditions on the inhomogeneity f assuring partial BMO-regularity for minima of quasiconvex functionals expressed in terms of the limiting function space L(n,∞).
In Section 2 we recall some well-known results from the study of nonlinear problems also establishing some Caccioppoli and Gehring type lemmas. In Section 3 we prove the excess decay estimates; considering separately the nondegenerate and the degenerate case. Section 4 is devoted to the proof of Theorem 1.1.
In (1.1), the integrand F:RN×n→R satisfies
{ F∈C2loc(RN×n) Λ−1|z|p≤F(z)≤Λ|z|p |∂2F(z)|≤Λ|z|p−2 |∂2F(z1)−∂2F(z2)|≤μ(|z2−z1||z2|+|z1|)(|z1|2+|z2|2)p−22 | (1.6) |
for all z∈RN×n, Λ≥1 being a positive absolute constant and μ:[0,∞)→[0,1] being a concave nondecreasing function with μ(0)=0. In the rest of the paper we will always assume p≥2. In order to derive meaningful regularity results, we need to update (1.5) to the stronger strict quasiconvexity condition
∫B[F(z+Dφ)−F(z)] dx≥λ∫B(|z|2+|Dφ|2)p−22|Dφ|2dx, | (1.7) |
holding for all z∈RN×n and φ∈W1,p0(B,RN), with λ being a positive, absolute constant. Furthermore, we allow the integrand F(⋅) to be degenerate elliptic in the origin. More specifically, we assume that F(⋅) features degeneracy of p-Laplacian type at the origin, i.e.,
| ∂F(z)−∂F(0)−|z|p−2z|z|p−1 |→0as|z|→0, | (1.8) |
which means that we can find a function ω:(0,∞)→(0,∞) such that
|z|≤ω(s) ⟹ |∂F(z)−∂F(0)−|z|p−2z|≤s|z|p−1, | (1.9) |
for every z∈RN×n and all s∈(0,∞). Moreover, the right-hand side term f:Ω→RN in (1.1) verifies as minimal integrability condition the following
f∈Lm(Ω,RN)with 2>m>{ 2n/(n+2)if n>2, 3/2if n=2, | (1.10) |
which, being p≥2, in turn implies that
f∈W1,p(Ω,RN)∗andm′<2∗≤p∗. | (1.11) |
Here it is intended that, when p≥n, the Sobolev conjugate exponent p∗ can be chosen as large as needed - in particular it will always be larger than p. By (1.5) and (1.6)2 we have
|∂F(z)|≤c|z|p−1, | (1.12) |
with c≡c(n,N,Λ,p); see for example [35, proof of Theorem 2.1]. Finally, (1.7) yields that for all z∈RN×n, ξ∈RN, ζ∈Rn it is
∂2F(z)⟨ξ⊗ζ,ξ⊗ζ⟩≥2λ|z|p−2|ξ|2|ζ|2, | (1.13) |
see [24, Chapter 5].
In this section we display our notation and collect some basic results that will be helpful later on.
In this paper, Ω⊂Rn is an open, bounded domain with Lipschitz boundary, and n≥2. By c we will always denote a general constant larger than one, possibly depending on the data of the problem. Special occurrences will be denoted by c∗,˜c or likewise. Noteworthy dependencies on parameters will be highlighted by putting them in parentheses. Moreover, to simplify the notation, we shall array the main parameters governing functional (1.1) in the shorthand data:=(n,N,λ,Λ,p,μ(⋅),ω(⋅)). By Br(x0):={x∈Rn:|x−x0|<r}, we denote the open ball with radius r, centred at x0; when not necessary or clear from the context, we shall omit denoting the center, i.e., Br(x0)≡Br - this will happen, for instance, when dealing with concentric balls. For x0∈Ω, we abbreviate dx0:=min{1,dist(x0,∂Ω)}. Moreover, with B⊂Rn being a measurable set with bounded positive Lebesgue measure 0<|B|<∞, and a:B→Rk, k≥1, being a measurable map, we denote
(a)B≡−∫Ba(x) dx:=1|B|∫Ba(x)dx. |
We will often employ the almost minimality property of the average, i.e.,
( −∫B|a−(a)B|tdx)1/t≤2( −∫B|a−z|tdx)1/t | (2.1) |
for all z∈RN×n and any t≥1. Finally, if t>1 we will indicate its conjugate by t′:=t/(t−1) and its Sobolev exponents as t∗:=nt/(n−t) if t<n or any number larger than one for t≥n and t∗:=max{nt/(n+t),1}.
When dealing with p-Laplacian type problems, we shall often use the auxiliary vector field Vs:RN×n→RN×n, defined by
Vs(z):=(s2+|z|2)(p−2)/4zwithp∈(1,∞), s≥0, z∈RN×n, |
incorporating the scaling features of the p-Laplacian. If s=0 we simply write Vs(⋅)≡V(⋅). A couple of useful related inequalities are
{ |Vs(z1)−Vs(z2)|≈(s2+|z1|2+|z2|2)(p−2)/4|z1−z2|, |Vs(z1+z2)|≲|Vs(z1)|+|Vs(z2)|, |Vs1(z)|≈|Vs2(z)|, if 12s2≤s1≤2s2, |V(z1)−V(z2)|2≈|V|z1|(z1−z2)|2, if 12|z2|≤|z1|≤2|z2|, | (2.2) |
and
|Vs(z)|2≈sp−2|z|2+|z|pwith p≥2, | (2.3) |
where the constants implicit in " ≲", " ≈" depend on n,N,p. A relevant property which is relevant for the nonlinear setting is recorded in the following lemma.
Lemma 2.1. Let t>−1, s∈[0,1] and z1,z2∈RN×n be such that s+|z1|+|z2|>0. Then
∫10[s2+|z1+y(z2−z1)|2]t2 dy≈(s2+|z1|2+|z2|2)t2, |
with constants implicit in "≈" depending only on n,N,t.
The following iteration lemma will be helpful throughout the rest of the paper; for a proof we refer the reader to [24, Lemma 6.1].
Lemma 2.2. Let h:[ϱ0,ϱ1]→R be a non-negative and bounded function, and let θ∈(0,1), A,B,γ1,γ2≥0 be numbers. Assume that h(t)≤θh(s)+A(s−t)−γ1+B(s−t)−γ2 holds for all ϱ0≤t<s≤ϱ1. Then the following inequality holds h(ϱ0)≤c(θ,γ1,γ2)[A(ϱ1−ϱ0)−γ1+B(ϱ1−ϱ0)−γ2].
We will often consider the "quadratic" version of the excess functional defined in (1.4), i.e.,
˜E(w,z0;Bϱ(x0)):=( −∫Bϱ(x0)|V(Dw)−z0|2dx)12. | (2.4) |
In the particular case z0=(Dw)Bϱ(x0) (z0=(V(Dw))Bϱ(x0), resp.) we shall simply write E(w,(Dw)Bϱ(x0);Bϱ(x0))≡E(w;Bϱ(x0)) (˜E(w,(V(Dw))Bϱ(x0);Bϱ(x0))≡˜E(w;Bϱ(x0)), resp.). A simple computation shows that
E(w;Bϱ(x0))p/2≈˜E(w;Bϱ(x0)). | (2.5) |
Moreover, from (2.1) and from [23, Formula (2.6)] we have that
˜E(w;Bϱ(x0))≈˜E(w,V((Dw)Bϱ(x0));Bϱ(x0)). | (2.6) |
In this section we collect some basic estimates for local minimizers of nonhomogeneous quasiconvex functionals. We start with a variation of the classical Caccioppoli inequality accounting for the presence of a nontrivial right-hand side term, coupled with an higher integrability result of Gehring-type.
Lemma 2.3. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let u∈W1,p(Ω,RN) be a local minimizer of functional (1.1).
● For every ball Bϱ(x0)⋐Ω and any u0∈RN, z0∈RN×n∖{0} it holds that
E(u,z0;Bϱ/2(x0))p≤c−∫Bϱ(x0)|z0|p−2|u−ℓϱ|2+|u−ℓϱ|pdx+c|z0|p−2(ϱm−∫Bϱ(x0)|f|m dx)2m, | (2.7) |
where E(⋅) is defined in (1.4), ℓ(x):=u0+⟨z0,x−x0⟩ and c≡c(n,N,λ,Λ,p).
● There exists an higher integrability exponent p2≡p2(n,N,λ,Λ,p)>p such that Du∈Lp2loc(Ω,RN×n) and the reverse Hölder inequality
( −∫Bϱ/2(x0)|Du−(Du)Bϱ(x0)|p2dx)1p2≤c( −∫Bϱ(x0)|Du|pdx)1p+c(ϱm−∫Bϱ(x0)|f|mdx)1m(p−1), | (2.8) |
is verified for all balls Bϱ(x0)⋐Ω with c≡c(n,N,λ,Λ,p).
Proof. For the ease of exposition, we split the proof in two steps, each of them corresponding to the proof of (2.7) and (2.8) respectively.
Step 1: proof of (2.7).
We choose parameters ϱ/2≤τ1<τ2≤ϱ, a cut-off function η∈C1c(Bτ2(x0)) such that 1Bτ1(x0)≤η≤1Bτ2(x0) and |Dη|≲(τ2−τ1)−1. Set φ1:=η(u−ℓ), φ2:=(1−η)(u−ℓ) and use (1.7) and the equivalence in (2.2)1 to estimate
c∫Bτ2(x0)|V|z0|(Dφ1)|2 dx≤∫Bτ2(x0)[F(z0+Dφ1)−F(z0)]dx=∫Bτ2(x0)[F(Du−Dφ2)−F(Du)]dx+∫Bτ2(x0)[F(Du)−F(Du−Dφ1)]dx+∫Bτ2(x0)[F(z0+Dφ2)−F(z0)]dx=:I1+I2+I3, | (2.9) |
where we have used the simple relation Dφ1+Dφ2=Du−z0. Terms I1 and I3 can be controlled as done in [19, Proposition 2]; indeed we have
I1+I3≤c∫Bτ2(x0)∖Bτ1(x0)|V|z0|(Dφ2)|2dx+c∫Bτ2(x0)∖Bτ1(x0)|V|z0|(Du−z0)|2dx(2.2)2≤c∫Bτ2(x0)∖Bτ1(x0)|V|z0|(Du−z0)|2+| V|z0|(u−ℓτ2−τ1) |2dx, | (2.10) |
for c≡c(n,N,λ,Λ,p). Concerning term I2, we exploit (1.10), the fact that φ1∈W1,p0(Bτ2(x0),RN) and apply Sobolev-Poincaré inequality to get
I2≤|Bτ2(x0)|(τm2−∫Bτ2(x0)|f|m dx)1/m(τ−m′2−∫Bτ2(x0)|φ1|m′ dx)1m′≤|Bτ2(x0)|(τm2−∫Bτ2(x0)|f|m dx)1/m( −∫Bτ2(x0)| φ1τ2 |2∗ dx)12∗≤|Bτ2(x0)|(τm2−∫Bτ2(x0)|f|mdx)1/m( −∫Bτ2(x0)|Dφ1|2dx)12≤ε∫Bτ2(x0)|V|z0|(Dφ1)|2dx+c|Bϱ(x0)|ε|z0|p−2(ϱm−∫Bϱ(x0)|f|mdx)2m, | (2.11) |
where c≡c(n,N,m) and we also used that ϱ/2≤τ2≤ϱ. Merging the content of the two above displays, recalling that η≡1 on Bτ1(x0) and choosing ε>0 sufficiently small, we obtain
∫Bτ1(x0)|V|z0|(Du−z0)|2dx≤c∫Bτ2(x0)∖Bτ1(x0)|V|z0|(Du−z0)|2+| V|z0|(u−ℓτ2−τ1) |2dx+c|Bϱ(x0)||z0|p−2(ϱm−∫Bϱ(x0)|f|mdx)2m, |
with c≡c(n,N,λ,Λ,p). At this stage, the classical hole-filling technique, Lemma 2.2 and (2.3) yield (2.7) and the first bound in the statement is proven.
Step 2: proof of (2.8).
To show the validity of (2.8), we follow [33, proof of Proposition 3.2] and first observe that if u is a local minimizer of functional F(⋅) on Bϱ(x0), setting fϱ(x):=ϱf(x0+ϱx), the map uϱ(x):=ϱ−1u(x0+ϱx) is a local minimizer on B1(0) of an integral with the same integrand appearing in (1.1) satisfying (1.6)1,2,3 and fϱ replacing f. This means that (2.10) still holds for all balls Bσ/2(˜x)⊆Bτ1(˜x)⊂Bτ2(˜x)⊆Bσ(˜x)⋐B1(0), with ˜x∈B1(0) being any point, in particular it remains true if |z0|=0, while condition |z0|≠0 was needed only in the estimate of term I2 in (2.11), that now requires some change. So, in the definition of the affine map ℓ we choose z0=0, u0=(uϱ)Bσ(˜x) and rearrange estimates (2.10) and (2.11) as:
I1+I3(2.3)≤c∫Bτ2(˜x)∖Bτ1(˜x)|Duϱ|p+| uϱ−(uϱ)Bσ(˜x)τ2−τ1 |pdx, |
and, recalling that φ1∈W1,p0(Bτ2(˜x),RN), via Sobolev Poincaré, Hölder and Young inequalities and (1.11)2, we estimate
I2≤|Bτ2(˜x)|(τ(p∗)′2−∫Bτ2(˜x)|fϱ|(p∗)′ dx)1(p∗)′(τ−p∗2−∫Bτ2(˜x)|φ1|p∗ dx)1p∗≤c|Bτ2(˜x)|(τ(p∗)′2−∫Bτ2(˜x)|fϱ|(p∗)′dx)1(p∗)′( −∫Bτ2(˜x)|Dφ1|pdx)1p≤c|Bσ(˜x)|ε1/(p−1)(σ(p∗)′−∫Bσ(˜x)|fϱ|(p∗)′dx)p(p∗)′(p−1)+ε∫Bτ2(˜x)|Dφ1|pdx, |
with c≡c(n,N,p). Plugging the content of the two previous displays in (2.9), reabsorbing terms and applying Lemma 2.2, we obtain
−∫Bσ/2(˜x)|Duϱ|p dx≤c −∫Bσ(˜x)| uϱ−(uϱ)Bσ(˜x)σ |p dx+c(σ(p∗)′−∫Bσ(˜x)|fϱ|(p∗)′dx)p(p∗)′(p−1), | (2.12) |
for c≡c(n,N,Λ,λ,p). Notice that
n(p(p∗)′(p−1)−1)≤pp−1, | (2.13) |
with equality holding when p<n, while for p≥n any value of p∗>1 will do. We then manipulate the second term on the right-hand side of (2.12) as
(σ(p∗)′−∫Bσ(˜x)|fϱ|(p∗)′dx)p(p∗)′(p−1)≤σpp−1−n(p(p∗)′(p−1)−1)( −∫B1(0)|fϱ|(p∗)′ dx)p(p∗)′(p−1)−1−∫Bσ(˜x)|fϱ|(p∗)′dx(2.13)≤( −∫B1(0)|fϱ|(p∗)′ dx)p(p∗)′(p−1)−1−∫Bσ(˜x)|fϱ|(p∗)′ dx=:−∫Bσ(˜x)|Kϱfϱ|(p∗)′ dx, |
where we set
K(p∗)′ϱ:=|B1(0)|1−p(p∗)′(p−1)‖fϱ‖pp−1−(p∗)′L(p∗)′(B1(0)). |
Plugging the content of the previous display in (2.12) and applying Sobolev-Poincaré inequality we get
−∫Bσ/2(˜x)|Duϱ|pdx≤c( −∫Bσ(˜x)|Duϱ|p∗ dx)pp∗+c −∫Bσ(˜x)|Kϱfϱ|(p∗)′ dx, |
with c≡c(n,N,Λ,λ,p). Now we can apply a variant of Gehring lemma [24, Corollary 6.1] to determine a higher integrability exponent s≡s(n,N,Λ,λ,p) such that 1<s≤m/(p∗)′ and
( −∫Bσ/2(˜x)|Duϱ|spdx)1sp≤c( −∫Bσ(˜x)|Duϱ|pdx)1p+cK(p∗)′/pϱ( −∫Bσ(˜x)|fϱ|s(p∗)′ dx)1sp |
for c≡c(n,N,Λ,λ,p). Next, notice that
K(p∗)′/pϱ=( −∫B1(0)|fϱ|(p∗)′dx)1(p∗)′(p−1)−1p≤( −∫B1(0)|fϱ|s(p∗)′dx)1s(p∗)′(p−1)−1sp, |
so plugging this last inequality in (2.14) and recalling that s(p∗)′≤m, we obtain
( −∫Bσ/2(˜x)|Duϱ|spdx)1sp≤c( −∫Bσ(˜x)|Duϱ|pdx)1p+c( −∫Bσ(˜x)|fϱ|m dx)1m(p−1). |
Setting p2:=sp>p above and recalling that ˜x∈B1(0) is arbitrary, we can fix ˜x=0, scale back to Bϱ(x0) and apply (2.1) to get (2.8) and the proof is complete.
In this section we prove some excess decay estimates considering separately two cases: when a smallness condition on the excess functional of our local minimizer u is satisfied and when such an estimate does not hold true.
We start working assuming that a suitable smallness condition on the excess functional E(u;Bϱ(x0)) is fulfilled. In particular, we prove the following proposition.
Proposition 3.1. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let u∈W1,p(Ω,RN) be a local minimizer of functional (1.1). Then, for τ0∈(0,2−10), there exists ε0≡ε0(data,τ0)∈(0,1) and ε1≡ε1(data,τ0)∈(0,1) such that the following implications hold true.
● If the conditions
E(u;Bϱ(x0))≤ε0|(Du)Bϱ(x0)|, | (3.1) |
and
(ϱm−∫Bϱ(x0)|f|m dx)1m≤ε1|(Du)Bϱ(x0)|p−22E(u;Bϱ(x0))p2, | (3.2) |
are verified on Bϱ(x0), then it holds that
E(u;Bτ0ϱ(x0))≤c0τβ00E(u;Bϱ(x0)), | (3.3) |
for all β0∈(0,2/p), with c0≡c0(data)>0.
● If condition (3.1) holds true and
(ϱm−∫Bϱ(x0)|f|m dx)1m>ε1|(Du)Bϱ(x0)|p−22E(u;Bϱ(x0))p2, | (3.4) |
is satisfied on Bϱ(x0), then
E(u;Bτ0ϱ(x0))≤c0(ϱm−∫Bϱ(x0)|f|m dx)1m(p−1), | (3.5) |
for c0≡c0(data)>0.
Proof of Proposition 3.1. For the sake of readability, since all balls considered here are concentric to Bϱ(x0), we will omit denoting the center. Moreover, we will adopt the following notation (Du)Bς(x0)≡(Du)ς and, for all φ∈C∞c(Bϱ;RN), we will denote ‖Dφ‖L∞(Bϱ)≡‖Dφ‖∞. We spilt the proof in two steps.
Step 1: proof of (3.3).
With no loss of generality we can assume that E(u;Bϱ)>0, which clearly implies, thanks to (3.1), that |(Du)ϱ|>0.
We begin proving that condition (3.1) implies that
−∫Bϱ|Du|pdx≤c|(Du)ϱ|p, | (3.6) |
for a constant c≡c(p,ε0)>0. Indeed,
−∫Bϱ|Du|pdx≤c −∫Bϱ|Du−(Du)ϱ|pdx+c|(Du)ϱ|p(1.4)≤cE(u;Bϱ)p+c|(Du)ϱ|p(3.1)≤c(εp0+1)|(Du)ϱ|p, |
and (3.6) follows.
Consider now
Bϱ∋x↦u0(x):=|(Du)ϱ|p−22(u(x)−(u)ϱ−⟨(Du)ϱ,x−x0⟩)E(u;Bϱ)p/2, | (3.7) |
and
d:=(E(u;Bϱ)|(Du)ϱ|)p2. |
Let us note that we have
−∫Bϱ|Du0|2dx+dp−2−∫Bϱ|Du0|pdx≤|(Du)ϱ|p−2E(u;Bϱ)p−∫Bϱ|Du−(Du)ϱ|2dx+(E(u;Bϱ)|(Du)ϱ|)p(p−2)2|(Du)ϱ|p(p−2)2E(u;Bϱ)p22−∫Bϱ|Du−(Du)ϱ|pdx≤1E(u;Bϱ)p−∫Bϱ|(Du)ϱ|p−2|Du−(Du)ϱ|2dx+1E(u;Bϱ)p−∫Bϱ|Du−(Du)ϱ|pdx≤1. |
Since |(Du)ϱ|>0 we have that the hypothesis of [12, Lemma 3.2] are satisfied with
A:=∂2F((Du)ϱ)|(Du)ϱ|2−p. | (3.8) |
Then,
| −∫BϱA⟨Du0,Dφ⟩ dx|≤c‖Dφ‖∞|(Du)ϱ|2−p2E(u;Bϱ)p2(ϱm−∫Bϱ|f|m dx)1m+c‖Dφ‖∞μ(E(u;Bϱ)|(Du)ϱ|)1p[1+(E(u;Bϱ)|(Du)ϱ|)p−22](3.1),(3.2)≤cε1‖Dφ‖∞+c‖Dφ‖∞μ(ε0)1p[1+εp−220]. |
Fix ε>0 and let δ≡δ(data,ε)>0 be the one given by [33, Lemma 2.4] and choose ε0 and ε1 sufficiently small such that
cε1+cμ(ε0)1p[1+εp−220]≤δ. | (3.9) |
With this choice of ε0 and ε1 it follows that u0 is almost A-harmonic on Bϱ, in the sense that
| −∫BϱA⟨Du0,Dφ⟩ dx|≤δ‖Dφ‖∞, |
with A as in (3.8). Hence, by [33, Lemma 2.4] we obtain that there exists h0∈W1,2(Bϱ;RN) which is A-harmonic, i.e.,
∫BϱA⟨Dh0,Dφ⟩ dx=0for all φ∈C∞c(Bϱ;RN), |
such that
−∫B3ϱ/4|Dh0|2dx+dp−2−∫B3ϱ/4|Dh0|pdx≤82np, | (3.10) |
and
−∫B3ϱ/4|u0−h0ϱ|2+dp−2|u0−h0ϱ|pdx≤ε. | (3.11) |
We choose now τ0∈(0,2−10), which will be fixed later on, and estimate
−∫B2τ0ϱ|u0(x)−h0(x0)−⟨Dh0(x0),x−x0⟩τ0ϱ|2dx≤c−∫B2τ0ϱ|h0(x)−h0(x0)−⟨Dh0(x0),x−x0⟩τ0ϱ|2dx+c−∫B2τ0ϱ|u0−h0τ0ϱ|2dx(3.11)≤c(τ0ϱ)2supBϱ/2|D2h0|2+cετn+20≤cτ20−∫B3ϱ/4|Dh0|2dx+cετn+20(3.10)≤cτ20+cετn+20, | (3.12) |
where c≡c(data)>0 and where we have used the following property of A-harmonic functions
ϱγsupBϱ/2|D2h0|γ≤c−∫B3ϱ/4|Dh0|γdx, | (3.13) |
with γ>1 and c depending on n, N, and on the ellipticity constants of A.
Now, choosing
ε:=τn+2p0, |
we have that this together with (3.9) gives that ε0≡ε0(data,τ0) and ε1≡ε1(data,τ0). Recalling the definition of u0 in (3.7) and (3.12) we eventually arrive at
−∫B2τ0ϱ|u−(u)ϱ−⟨(Du)ϱ,x−x0⟩−|(Du)ϱ|2−p2E(u;Bϱ)p/2(h0(x0)−⟨Dh0(x0),x−x0⟩)|2(τ0ϱ)2dx≤c|(Du)ϱ|2−pE(u;Bϱ)pτ20, | (3.14) |
for c≡c(data)>0. By a similar computation, always using (3.13), (3.10) and (3.11), we obtain that
dp−2−∫B2τ0ϱ|u0−h0(x0)−⟨Dh0(x0),x−x0⟩τ0ϱ|pdx≤cdp−2(τ0ϱ)psupBϱ/2|D2h0|p+cετn+p0≤cτp0. |
In this way, as for (3.14), by the definition of u0 in (3.7), we eventually arrive at
−∫B2τ0ϱ|u−(u)ϱ−⟨(Du)ϱ,x−x0⟩−|(Du)ϱ|2−p2E(u;Bϱ)p/2(h0(x0)−⟨Dh0(x0),x−x0⟩)|p(τ0ϱ)pdx≤cd2−p|(Du)ϱ|p(2−p)2E(u;Bϱ)p22τp0≤cE(u;Bϱ)pτ20, | (3.15) |
with c≡c(data).
Denote now with ℓ2τ0ϱ the unique affine function such that
ℓ2τ0ϱ↦minℓ affine−∫B2τ0ϱ|u−ℓ|2dx. |
Hence, by (3.14) and (3.15), we conclude that
−∫B2τ0ϱ|(Du)ϱ|p−2|u−ℓ2τ0ϱ2τ0ϱ|2+|u−ℓ2τ0ϱ2τ0ϱ|pdx≤cτ2E(u;Bϱ)p. | (3.16) |
Notice that we have also used the property that
−∫Bϱ|u−ℓϱ|pdx≤c−∫Bϱ|u−ℓ|pdx, |
for p≥2, c≡c(n,N,p)>0 and for any affine function ℓ; see [33, Lemma 2.3].
Recalling the definition of the excess functional E(⋅), in (1.4), we can estimate the following quantity as follows
|Dℓ2τ0ϱ−(Du)ϱ|≤|Dℓ2τ0ϱ−(Du)2τ0ϱ|+|(Du)2τ0ϱ−(Du)ϱ|≤c(−∫B2τ0ϱ|Du−(Du)2τ0ϱ|2dx)12+(−∫B2τ0ϱ|Du−(Du)ϱ|2dx)12(2.1)≤cτn/20(−∫Bϱ|Du−(Du)ϱ|2dx)12=c|(Du)ϱ|2−p2τn/20(−∫Bϱ|(Du)ϱ|p−2|Du−(Du)ϱ|2dx)12≤c(n)τn/20(E(u,Bϱ)|(Du)ϱ|)p2|(Du)ϱ|, | (3.17) |
where we have used the following property of the affine function ℓ2τ0ϱ
|Dℓ2τ0ϱ−(Du)2τ0ϱ|p≤c−∫B2τ0ϱ|Du−(Du)2τ0ϱ|pdx, |
for a constant c≡c(n,p)>0; see for example [33, Lemma 2.2].
Now, starting from (3.1) and (3.9), we further reduce the size of ε0 such that
(E(u,Bϱ)|(Du)ϱ|)p2(3.1)≤εp20≤τn/208c(n), | (3.18) |
where c≡c(n) is the same constant appearing in (3.17). Thus, combining (3.17) and (3.18), we get
|Dℓ2τ0ϱ−(Du)ϱ|≤|(Du)ϱ|8. | (3.19) |
The information provided by (3.18) combined with (3.16) allow us to conclude that
−∫B2τ0ϱ|Dℓ2τ0ϱ|p−2|u−ℓ2τ0ϱ2τ0ϱ|2+|u−ℓ2τ0ϱ2τ0ϱ|pdx≤cτ2E(u;Bϱ)p. | (3.20) |
By triangular inequality and (3.19) we also get
|Dℓ2τ0ϱ|≥|(Du)ϱ|−|Dℓ2τ0ϱ−(Du)ϱ|(3.19)≥7|(Du)ϱ|8 |
which, therefore, implies that
−∫Bτ0ϱ|Dℓ2τ0ϱ|p−2|Du−Dℓ2τ0ϱ|2dx+infz∈RN×n−∫Bτ0ϱ|Du−z|pdx(2.7)≤c−∫B2τ0ϱ|Dℓ2τ0ϱ|p−2|u−ℓ2τ0ϱ2τ0ϱ|2+|u−ℓ2τ0ϱ2τ0ϱ|pdx+c|Dℓ2τ0ϱ|p−2((2τ0ϱ)m−∫B2τ0ϱ|f|mdx)2m(3.20)≤cτ20E(u,Bϱ)p+cτ2−2n/m0|(Du)ϱ|p−2(ϱm−∫Bϱ|f|mdx)2m, | (3.21) |
where c≡c(data)>0. By triangular inequality, we can further estimate
−∫Bτ0ϱ|(Du)τ0ϱ|p−2|Du−(Du)τ0ϱ|2dx≤c−∫Bτ0ϱ|Dℓτ0ϱ−(Du)τ0ϱ|p−2|Du−(Du)τ0ϱ|2dx+c−∫Bτ0ϱ|Dℓ2τ0ϱ−Dℓτ0ϱ|p−2|Du−(Du)τ0ϱ|2dx+c−∫Bτ0ϱ|Dℓ2τ0ϱ|p−2|Du−(Du)τ0ϱ|2dx=I1+I2+I3, |
where c≡c(p)>0. We now separately estimate the previous integrals. We begin considering I1. By Young and triangular inequalities we get
I1≤c|Dℓτ0ϱ−(Du)τ0ϱ|p+c−∫Bτ0ϱ|Du−(Du)τ0ϱ|pdx≤c−∫Bτ0ϱ|Du−(Du)τ0ϱ|pdx(2.1)≤cinfz∈RN−∫Bτ0ϱ|Du−z|pdx(3.21)≤cτ20E(u,Bϱ)p+cτ2−2n/m0|(Du)ϱ|p−2(ϱm−∫Bϱ|f|mdx)2m, |
with c≡c(data)>0. In a similar fashion, we can treat the integral I2
I2≤c|Dℓ2τ0ϱ−Dℓτ0ϱ|p+c−∫Bτ0ϱ|Du−(Du)τ0ϱ|pdx(2.1)≤c−∫B2τ0ϱ|u−ℓ2τ0ϱ2τ0ϱ|pdx+cinfz∈RN×n−∫Bτ0ϱ|Du−z|pdx(3.20),(3.21)≤cτ20E(u,Bϱ)p+cτ2−2n/m0|(Du)ϱ|p−2(ϱm−∫Bϱ|f|mdx)2m, |
where we have used the following property of the affine function ℓ2τ0ϱ
|Dℓ2τ0ϱ−Dℓτ0ϱ|p≤c−∫B2τ0ϱ|u−ℓ2τ0ϱ2τ0ϱ|pdx, |
for a given constant c≡c(n,p)>0; see [33, Lemma 2.2]. Finally, the last integral I3 can be treated recalling (3.21) and (2.1), i.e.,
I3≤cτ20E(u,Bϱ)p+cτ2−2n/m0|(Du)ϱ|p−2(ϱm−∫Bϱ|f|mdx)2m. |
All in all, combining the previous estimate
E(u;Bτ0ϱ)≤cτ2/p0E(u,Bϱ)+cτ2/p−2n/(mp)0|(Du)ϱ|p−2p(ϱm−∫Bϱ|f|mdx)2mp(3.2)≤cτ2/p0E(u,Bϱ)+cτ2/p−2n/(mp)0ε2/p1E(u;Bτ0ϱ)≤c0τ2/p0E(u;Bτ0ϱ), |
up to choosing ε1 such that
ε1≤τn/m0. |
Step 2: proof of (3.5).
The proof follows by [12, Lemma 2.4] which yields
E(u;Bτ0ϱ(x0))p2≤23pτn/20E(u;Bϱ(x0))p2(3.4)≤23pτn/20ε−11|(Du)Bϱ(x0)|2−p2(ϱm−∫Bϱ(x0)|f|m dx)1m(3.1)≤26(p−1)τn(p−1)/p0εp−220ε−11E(u;Bτ0ϱ)2−p2(ϱm−∫Bϱ(x0)|f|m dx)1m. |
Multiplying both sides by E(u;Bτ0ϱ)p−22 we get the desired estimate.
It remains to considering the case when condition (3.1) does not hold true. We start with two technical lemmas. The first one is an analogous of the Caccioppoli inequality (2.7), where we take in consideration the eventuality z0=0.
Lemma 3.1. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let u∈W1,p(Ω,RN) be a local minimizer of functional (1.1). For every ball Bϱ(x0)⋐Ω and any u0∈RN, z0∈RN×n it holds that
E(u,z0;Bϱ/2(x0))p≤c −∫Bϱ(x0)|z0|p−2|u−ℓϱ|2+|u−ℓϱ|p dx+c(ϱm−∫Bϱ(x0)|f|m dx)pm(p−1), | (3.22) |
where E(⋅) is defined in (1.4), ℓ(x):=u0+⟨z0,x−x0⟩ and c≡c(n,N,λ,Λ,p).
Proof. The proof is analogous to estimate (2.7), up to treating in a different way the term I2 in (2.9), taking in consideration the eventuality z0=0. Exploiting (1.10) and fact that φ1∈W1,p0(Bτ2(x0),RN), an application of the Sobolev-Poincaré inequality yields
I2≤|Bτ2(x0)|(τm2−∫Bτ2(x0)|f|m dx)1/m(τ−m′2−∫Bτ2(x0)|φ1|m′ dx)1m′≤|Bτ2(x0)|(τm2−∫Bτ2(x0)|f|m dx)1/m( −∫Bτ2(x0)| φ1τ2 |p∗ dx)1p∗≤|Bτ2(x0)|(τm2−∫Bτ2(x0)|f|mdx)1/m( −∫Bτ2(x0)|Dφ1|pdx)1p≤ε∫Bτ2(x0)|V|z0|(Dφ1)|2dx+c|Bϱ(x0)|ε1/(p−1)(ϱm−∫Bϱ(x0)|f|mdx)pm(p−1), | (3.23) |
where c≡c(n,N,m) and we also used that ϱ/2≤τ2≤ϱ. Hence, proceeding as in the proof of (2.7), we obtain that
∫Bτ1(x0)|V|z0|(Du−z0)|2dx≤c∫Bτ2(x0)∖Bτ1(x0)|V|z0|(Du−z0)|2+| V|z0|(u−ℓτ2−τ1) |2dx+c|Bϱ(x0)|ε1/(p−1)(ϱm−∫Bϱ(x0)|f|mdx)pm(p−1), |
with c≡c(n,N,λ,Λ,p). Concluding as in the proof of (2.7), we eventually arrive at (3.22).
Lemma 3.2. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let u∈W1,p(Ω,RN) be a local minimizer of functional (1.1). For any Bϱ(x0)⋐Ω and any s∈(0,∞) it holds that
| −∫Bϱ(x0)⟨|Du|p−2Du,Dφ⟩ dx|≤s‖Dφ‖L∞(Bϱ(x0))( −∫Bϱ(x0)|Du|p dx)p−1p+cω(s)−1‖Dφ‖L∞(Bϱ(x0))−∫Bϱ(x0)|Du|p dx+c‖Dφ‖L∞(Bϱ(x0))(ϱm−∫Bϱ(x0)|f|m dx)1/m, | (3.24) |
for any φ∈C∞0(Bϱ(x0),RN), with c≡c(n,N,Λ,λ,p).
Proof. Given the regularity properties of the integrand F, we have that a local minimizer u of (1.1) solves weakly the following integral identity (see [42, Lemma 7.3])
∫Ω[⟨∂F(Du),Dφ⟩−f⋅φ] dx=0for all φ∈C∞0(Ω,RN). | (3.25) |
Now, fix φ∈C∞0(Bϱ(x0),RN) and split
| −∫Bϱ(x0)⟨|Du|p−2Du,Dφ⟩ dx|(3.25)≤| −∫Bϱ(x0)⟨∂F(Du)−∂F(0)−|Du|p−2Du,Dφ⟩ dx|+| −∫Bϱ(x0)f⋅φ dx|=:I1+I2. |
We begin estimating the first integral I1. For s∈(0,∞) we get
I1≤‖Dφ‖L∞(Bϱ(x0))|Bϱ(x0)|∫Bϱ(x0)∩{|Du|≤ω(s)}|∂F(Du)−∂F(0)−|Du|p−2Du| dx+‖Dφ‖L∞(Bϱ(x0))|Bϱ(x0)|∫Bϱ(x0)∩{|Du|>ω(s)}|∂F(Du)−∂F(0)−|Du|p−2Du| dx≤s‖Dφ‖L∞(Bϱ(x0))( −∫Bϱ(x0)|Du|p dx)p−1p+cω(s)−1‖Dφ‖L∞(Bϱ(x0))−∫Bϱ(x0)|Du|p dx. | (3.26) |
On the other hand, the integral I2 can be estimated as follows
I2≤(ϱm−∫Bϱ(x0)|f|m dx)1/m( −∫Bϱ(x0)|φϱ|m′ dx)1m′≤(ϱm−∫Bϱ(x0)|f|m dx)1/m( −∫Bϱ(x0)| φϱ |p∗ dx)1p∗≤(ϱm−∫Bϱ(x0)|f|mdx)1/m( −∫Bϱ(x0)|Dφ|pdx)1p≤‖Dφ‖L∞(Bϱ(x0))(ϱm−∫Bϱ(x0)|f|mdx)1/m. |
Combining the inequalities above we obtain (3.24).
In this setting the analogous result of Proposition 3.1 is the following one.
Proposition 3.2. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let u∈W1,p(Ω,RN) be a local minimizer of functional (1.1). Then, for any χ∈(0,1] and any τ1∈(0,2−10), there exists ε2≡ε2(data,χ,τ1)∈(0,1) such that if the smallness conditions
χ|(Du)Bϱ(x0)|≤E(u;Bϱ(x0)),andE(u;Bϱ(x0))≤ε2, | (3.27) |
are satisfied on a ball Bϱ(x0)⊂Rn, then
E(u;Bτ1ϱ(x0))≤c1τβ11E(u;Bϱ(x0))+c1(ϱm−∫Bϱ(x0)|f|m dx)1m(p−1), | (3.28) |
for any β1∈(0,2α/p), with α≡α(n,N,p)∈(0,1) is the exponent in (3.34), and c1≡c1(data,χ).
Proof. We adopt the same notations used in the proof of Proposition 3.1. Let us begin noticing that condition (3.27)1 implies the following estimate
−∫Bϱ|Du|p dx≤cχE(u;Bϱ)pwithcχ:=2p(1+χ−p). | (3.29) |
Indeed, by (1.4) and (3.27), we have
−∫Bϱ|Du|p dx≤2p−∫Bϱ|Du−(Du)Bϱ|p dx+2p|(Du)Bϱ|p≤2pE(u;Bϱ)p+2pχpE(u;Bϱ)p. |
Consider now
κ:=cχE(u;Bϱ)+((ϱε3)m−∫Bϱ|f|mdx)1m(p−1)andv0:=uκ, |
for ε3∈(0,1], which will be fixed later on. Applying (3.24) to the function v0 yields
| −∫Bϱ/2(x0)⟨|Dv0|p−2Dv0,Dφ⟩dx|(3.27)2,(3.29)≤c‖Dφ‖∞(s+ω(s)−1ε2+ε3). |
For any ε>0 and ϑ∈(0,1) and let δ be the one given by [17, Lemma 1.1]. Then, up to choosing s, ε2 and ε3 sufficiently small, we arrive at
c(s+ω(s)−1ε2+ε3)≤δ‖Dφ‖p−1∞. |
Then, Lemma 1.1 in [17] implies
( −∫Bϱ/2|V(Dv0)−V(Dh)|2ϑdx)1ϑ≤cε −∫Bϱ/2|Du|pdx(3.29),(3.27)2≤cεεp2, |
up to taking ε as small as needed. Now, denoting with h0:=hκ, we have that
( −∫Bϱ/2|V(Du)−V(Dh0)|2ϑdx)1ϑ≤εεp2κp. |
Now, we choose ϑ:=(s)′/2, with s being the exponent given by (2.8). Note that by the proof of (2.8) it actually follows that ϑ<1. Thus, choosing εεp2κp≤τ2n+4α1 (where α∈(0,1) is given by (3.34)) we arrive at
( −∫Bϱ/2|V(Du)−V(Dh0)|(s)′dx)1(s)′≤cτn+2α1. |
By Hölder's Inequality, we have that
−∫Bϱ/2|V(Du)−V(Dh0)|2dx≤( −∫Bϱ/2|V(Du)−V(Dh0)|(s)′dx)1(s)′( −∫Bϱ/2|V(Du)−V(Dh0)|sdx)1s. | (3.30) |
Hence, since by (2.3) V(z)≈|z|p, an application of estimates (2.8) and (3.29) now yields
( −∫Bϱ/2|V(Du)|sdx)1s≤c( −∫Bϱ/2|Du−(Du)ϱ|p2dx)pp2+c|(Du)ϱ|p≤c −∫Bϱ|Du|pdx+c(ϱm−∫Bϱ|f|mdx)pm(p−1)+c|(Du)ϱ|p≤cE(u;Bϱ)p+c(ϱm−∫Bϱ|f|mdx)pm(p−1), | (3.31) |
with c≡c(data,χ).
On the other hand, by classical properties of p-harmonic functions, we have that
( −∫Bϱ/2|V(Dh0)|sdx)1s≤c −∫Bϱ|Dh0|pdx≤c −∫Bϱ|Du|pdx≤cE(u;Bϱ)p. | (3.32) |
Hence, combining (3.30)–(3.32), we get that
−∫Bϱ/2|V(Du)−V(Dh0)|2dx≤cτn+2α1E(u;Bϱ)p+cτn+2α1(ϱm−∫Bϱ|f|mdx)pm(p−1). | (3.33) |
Let us recall that, for any τ1∈(0,2−10), given the p-harmonic function h0 we have
˜E(h0;Bτ1ϱ)2≤cτ2α1κp,α≡α(n,N,p)∈(0,1). | (3.34) |
Moreover, using Jensen's Inequality we can estimate the following difference as follows
|(Du)τ1ϱ−(Du)ϱ|≤( −∫Bτ1ϱ|Du−(Du)ϱ|pdx)1p≤τ−np1( −∫Bϱ|Du−(Du)ϱ|pdx)1p(1.4),(3.27)2≤τ−np1ε2. |
Thus, up to taking ε2 sufficiently small, by the triangular inequality, we obtain that
12|(Du)τ1ϱ|≤|(Du)ϱ|≤2|(Du)τ1ϱ|. |
Hence, (2.2) yield
|V|(Du)τ1ϱ|(⋅)|2≈|V|(Du)ϱ|(⋅)|2, |
and
|V((Du)τ1ϱ)−V((Du)ϱ)|2≈|V|(Du)ϱ|((Du)ϱ−(Du)τ1ϱ)|2. |
Then,
E(u;Bτ1ϱ)p(2.5)≤c˜E(u;Bτ1ϱ)2(2.6)≤c−∫Bτ1ϱ|V(Du)−V((Du)τ1ϱ)|2dx≤cτ−n1 −∫Bϱ/2|V(Du)−V(Dh0)|2dx+c −∫Bτ1ϱ|V(Dh0)−V((Dh0)τ1ϱ)|2dx(2.6)≤cτ−n1 −∫Bϱ/2|V(Du)−V(Dh0)|2dx+c ˜E(h0,Bτ1ϱ)(3.33),(3.34)≤cτ2α1E(u;Bϱ)p+c(ϱm−∫Bϱ|f|mdx)pm(p−1), |
and the desired estimate (3.28) follows.
This section is devoted to the proof of Theorem 1.1. First, we prove the following proposition.
Proposition 4.1. Under assumptions (1.6)1,2,3, (1.7) and (1.10), let u∈W1,p(Ω,RN) be a local minimizer of functional (1.1). Then, there exists ε∗≡ε∗(data)>0 such that if the following condition
E(Du;Br)+supϱ≤r(ϱm−∫Bϱ|f|mdx)1m(p−1)<ε, | (4.1) |
is satisfied on Br⊂Ω, for some ε∈(0,ε∗], then
supϱ≤rE(Du;Bϱ)<c3ε, | (4.2) |
for c3≡c3(data)>0.
Proof. For the sake of readability, since all balls considered in the proof are concentric to Br(x0), we will omit denoting the center.
Let us start fixing an exponent β≡β(α,p) such that
0<β<min{β0,β1}=:βm, | (4.3) |
where β0 and β1 are the exponents appearing in Propositions 3.1 and 3.2. Moreover, given the constant c0 and c1 from Propositions 3.1 and 3.2, choose τ≡τ(data,β) such that
(c0+c1)τβm−β≤14. | (4.4) |
With the choice of τ0 as in (4.4) above, we can determine the constant ε0 and ε1 of Proposition 3.1. Now, we proceed applying Proposition 3.2 taking χ≡ε0 and τ1 as in (4.4) there. This determines the constant ε2 and c2. We consider a ball Br⊂Ω such that
E(Du;Br)<ε2, | (4.5) |
and
supϱ≤rc2(ϱm−∫Bϱ|f|mdx)1m(p−1)≤ε24, | (4.6) |
where the constant c2:=c1+c0, with c0 appearing in (3.5) and c1 in (3.28). In particular, see that by (4.5) and (4.6) we are in the case when (4.1) does hold true.
Now, we recall Proposition 3.2. Seeing that (3.27)2 is satisfied (being (4.5)) we only check whether (3.27)1 is verified too. If ε0|(Du)Br|≤E(Du;Br) is satisfied then we obtain from (3.28), with τ1≡τ in (4.4) that
E(u;Bτr)≤τβ4E(u;Br)+c2(rm−∫Br|f|m dx)1m(p−1)≤τβ4E(u;Br)+supϱ≤rc2(ϱm−∫Bϱ|f|m dx)1m(p−1)≤τβ4E(u;Br)+ε24≤ε2, | (4.7) |
where the last inequality follows from (4.5) and (4.6). If on the other hand it holds ε0|(Du)Br|≥E(Du;Br), by Proposition 3.1, then by (3.3) or (3.5) we eventually arrive at the same estimate (4.7).
Iterating now the seam argument we arrive at
E(Du;Bτjr)<ε2for any j≥0, |
and the estimate
E(u;Bτj+1r)≤τβ4E(u;Bτjr)+c2((τjr)m−∫Bτjr|f|m dx)1m(p−1), |
holds true. By the inequality above we have that for any k≥0
E(u;Bτk+1r)≤τβ(k+1)4E(u;Br)+c2k∑j=0(τβ)j−k((τjr)m−∫Bτjr|f|m dx)1m(p−1)≤τβ(k+1)E(u;Br)+c2supϱ≤r(ϱm−∫Brr|f|mdx)1m(p−1). |
Applying a standard interpolation argument we conclude that, for any t≤r, it holds
E(Du,Bs)≤c3(sr)βE(Du,Br)+c3supϱ≤r(ϱm−∫Brr|f|mdx)1m(p−1), | (4.8) |
where c3≡c3(data). The desired estimate (4.2) now follows.
Proof of Theorem 1.1. We proceed following the same argument used in [33, Theorem 1.5]. We star proving that, for any 1≤m<n and any O⊂Ω, with positive measure, we have that
‖f‖Lm(O)≤(nn−m)1/m|O|1/m−1/n‖f‖Ln,∞(O). | (4.9) |
Indeed, fix ˉλ which will be chosen later on. Then, we have that
‖f‖mLm(O)=m∫ˉλ0λm|{x∈O:|f|>λ}|dλλ+m∫∞ˉλλm|{x∈O:|f|>λ}|dλλ. | (4.10) |
The first integral on the righthand side of (4.10) can be estimated in the following way
∫ˉλ0λm|{x∈O:|f|>λ}|dλλ≤ˉλm|O|m. |
On the other hand, the second integral can be estimated recalling the definition of the Ln,∞(O)-norm. Indeed,
∫∞ˉλλm|{x∈O:|f|>λ}|dλλ≤‖f‖nLn,∞(O)∫∞ˉλdλλ1+n−m≤‖f‖nLn,∞(O)(n−m)ˉλn−m. |
Hence, putting all the estimates above in (4.10), choosing ˉλ:=‖f‖Ln,∞(O)/|O|1/n, we obtain (4.9).
Now, recalling condition (1.2) we have that
(ϱm−∫Bϱ|f|mdx)1/m≤(nn−m)1/m|B1|−1/n‖f‖Ln,∞(Ω)(1.10)≤(4n/m|B1|)1/n‖f‖Ln,∞(Ω)(1.2)≤ε∗, |
where ε∗ is the one obtained in the proof of Proposition 4.1. From this it follows that, we can choose a radius ϱ1 such that
supϱ≤ϱ1c2(ϱm−∫Bϱ(x)|f|mdx)1/m(p−1)≤ε∗4c3. | (4.11) |
We want to show that the set Ωu appearing in (1.3) can be characterized by
Ωu:={x0∈Ω:∃Bϱ(x0)⋐Ωwithϱ≤ϱ1:E(Du,Bϱ(x0))<ε∗/(4c3)}, |
thus fixing ϱx0:=ϱ1 and εx0:=ε∗/(4c3). We first star noting that the the set Ωu defined in (1.4) is such that |ΩsetminusΩu|=0. Indeed, let us consider the set
Lu:={x0∈Ω:lim infϱ→0˜E(u;Bϱ(x0))2=0}, | (4.12) |
which is such that |ΩsetminusLu|=0 by standard Lebesgue's Theory. Moreover, by (2.5) it follows that
Lu:={x0∈Ω:lim infϱ→0E(u;Bϱ(x0))=0}, |
so that, Lu⊂Ωu and we eventually obtained that |ΩsetminusΩu|=0. Now we show that Ωu is open. Let us fix x0∈Ωu and find a radius ϱx0≤ϱ1 such that
E(Du,Bϱx0(x0))<ε∗4c3. | (4.13) |
By absolute continuity of the functional E(⋅) we have that there exists an open neighbourhood O(x0) such that, for any x∈O(x0) it holds
E(Du,Bϱx0(x))<ε∗4c3andBϱx0(x)⋐Ω. | (4.14) |
This prove that Ωu is open. Now let us start noting that (4.11) and (4.14) yield that condition (4.1) is satisfied with Br≡Bϱx0(x). Hence, an application of Proposition 4.1 yields
supt≤ϱx0E(Du,Bt(x))<ε∗, |
for any x∈O(x0). Thus concluding the proof.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The author is supported by INdAM Projects "Fenomeni non locali in problemi locali", CUP_E55F22000270001 and "Problemi non locali: teoria cinetica e non uniforme ellitticità", CUP_E53C220019320001, and also by the Project "Local vs Nonlocal: mixed type operators and nonuniform ellipticity", CUP_D91B21005370003.
The author declares no conflict of interest.
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