This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.
Citation: Yang Liu. Long-time behavior of a class of viscoelastic plate equations[J]. Electronic Research Archive, 2020, 28(1): 311-326. doi: 10.3934/era.2020018
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This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.
We consider the general second order elliptic equation in divergence form
n∑i=1∂∂xiai(x,u(x),Du(x))=b(x,u(x),Du(x)),x∈Ω, | (1.1) |
where Ω is an open set of Rn, n≥2, the vector field (ai(x,u,ξ))i=1,…,n and the right hand side b(x,u,ξ) are Carathéodory applications defined in Ω×R×Rn. We study the elliptic equations (1.1) under some general growth conditions on the gradient variable ξ=Du, named p,q− conditions, which we are going to state in the next Section 3.2. Under these assumptions we will obtain the local boundedness of the weak solutions, as stated in Theorem 3.2.
A strong motivation to study the local boundedness of solutions to (1.1) relies on the recent research in [53], where the local Lipschitz continuity of the weak solutions of the Eq (1.1) has been obtained under general growth conditions, precisely some p,q−growth assumptions, with the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable. In [53] the Sobolev class of functions where to start in order to get more regularity of the weak solutions was pointed out, precisely u∈W1,qloc(Ω)∩L∞loc(Ω). That is, in particular the local boundedness u∈L∞loc(Ω) of weak solutions is a starting assumption for more interior regularity; i.e., for obtaining u∈W1,∞loc(Ω) and more. When we refer to the classical cases this is a well known aspect which appears in the mathematical literature on a-priori regularity: in fact, for instance, under the so-called natural growth conditions, i.e., when q=p, then the a-priori boundedness of u often is a natural assumption to obtain the boundedness of its gradient Du too; see for instance the classical reference book by Ladyzhenskaya-Ural'tseva [45,Chapter 4,Section 3] and the C1,α−regularity result by Tolksdorf [60].
The aim of this paper is to derive the local boundedness of solutions to (1.1); i.e., to deduce the local boundedness of u only from the growth assumptions on the vector field (ai(x,u,ξ))i=1,…,n and the right hand side b(x,u,ξ) in (1.1). The precise conditions and the related results are stated in Section 3.
We start with a relevant aspect to remark in our context, which is different from what happens in minimization problems and it is peculiar for equations: although under p,q−growth conditions (with p<q) the Eq (1.1) is elliptic and coercive in W1,ploc(Ω), it is not possible a-priori to look for weak solutions only in the Sobolev class W1,ploc(Ω), but it is necessary to emphasize that the notion of weak solution is consistent if a-priori we assume u∈W1,qloc(Ω). This is detailed in Section 2.
Going into more detail, in this article we study the local boundedness of weak solutions to the p−elliptic equation (1.1) with q−growth, 1<p≤q<p+1, as in (3.2), (3.3) and (3.7)–(3.10). Starting from the integrability condition u∈W1,qloc(Ω) on the weak solution, under the bound on the ratio qp
qp<1+1n−1 |
we obtain u∈L∞loc(Ω). The proof is based on the powerful De Giorgi technique [29], by showing first a Caccioppoli-type inequality and then applying an iteration procedure. The result is obtained via a Sobolev embedding theorem on spheres, a procedure introduced by Bella and Schäffner in [3], that allows a dimensional gain in the gap between p and q. This idea has been later used by the same authors in [4], by Schäffner [58] and, particularly close to the topic of our paper, by Hirsch and Schäffner [43] and De Rosa and Grimaldi [30], where the local boundedness of scalar minimizers of a class of convex energy integrals with p,q−growth was obtained with the bound qp<1+qn−1.
Some references about the local boundedness of solutions to elliptic equations and systems, with general and p,q−growth conditions, start by Kolodīĭ [44] in 1970 in the specific case of some anisotropic elliptic equations. The local boundedness of solution to classes of anisotropic elliptic equations or systems have been investigated by the authors [18,19,20,21,22,23,24] and by Di Benedetto, Gianazza and Vespri [31]. Other results on the boundedness of solutions of PDEs or of minimizers of integral functionals can be found in Boccardo, Marcellini and Sbordone [7], Fusco and Sbordone [37,38], Stroffolini [59], Cianchi [14], Pucci and Servadei [57], Cupini, Leonetti and Mascolo [17], Carozza, Gao, Giova and Leonetti [12], Granucci and Randolfi [42], Biagi, Cupini and Mascolo [5].
Interior L∞−gradient bound, i.e., the local Lipschitz continuity, of weak solutions to nonlinear elliptic equations and systems under non standard growth conditions have been obtained since 1989 in [46,47,48,49,50]. See also the following recent references for other Lipschitz regularity results: Colombo and Mingione [16], Baroni, Colombo and Mingione [1], Eleuteri, Marcellini and Mascolo [34,35], Di Marco and Marcellini [32], Beck and Mingione [2], Bousquet and Brasco [9], De Filippis and Mingione [26,27], Caselli, Eleuteri and Passarelli di Napoli [13], Gentile [39], the authors and Passarelli di Napoli [25], Eleuteri, Marcellini, Mascolo and Perrotta [36]; see also [53]. For other related results see also Byun and Oh [10] and Mingione and Palatucci [55]. The local boundedness of the solution u can be used to achieve further regularity properties, as the Hölder continuity of u or of its gradient Du; we limit here to cite Bildhauer and Fuchs [6], Düzgun, Marcellini and Vespri [33], Di Benedetto, Gianazza and Vespri [31], Byun and Oh [11] as examples of this approach. For recent boundary regularity results in the context considered in this manuscript we mention Cianchi and Maz'ya [15], Bögelein, Duzaar, Marcellini and Scheven [8], De Filippis and Piccinini [28]. A well known reference about the regularity theory is the article [54] by Giuseppe Mingione. We also refer to [51,52,53] and to De Filippis and Mingione [27], Mingione and Rădulescu [56], who have outlined the recent trends and advances in the regularity theory for variational problems with non-standard growths and non-uniform ellipticity.
In order to investigate the consistency of the notion of weak solution, we anticipate the ellipticity and growth conditions of Section 3, in particular the growth in (3.3), (3.4),
{|ai(x,u,ξ)|≤Λ{|ξ|q−1+|u|γ1+b1(x)},∀i=1,…,n,|b(x,u,ξ)|≤Λ{|ξ|r+|u|γ2+b2(x)}. | (2.1) |
As well known the integral form of the equation, for a smooth test function φ with compact support in Ω, is
∫Ωn∑i=1ai(x,u,Du)φxidx+∫Ωb(x,u,Du)φdx=0. |
Let us discuss the summability conditions for the pairings above to be well defined. Since each ai in the gradient variable ξ grows at most as |ξ|q−1, more generally we can consider test functions φ∈W1,q0(Ω). In fact, starting with the first addendum and applying the Young inequality with conjugate exponents qq−1 and q, we obtain the L1 local summability
|ai(x,u,Du)φxi|≤Λ{|Du|q−1+|u|γ1+b1(x)}|φxi|≤Λq−1q{|Du|q−1+|u|γ1+b1(x)}qq−1+Λq|φxi|q∈L1loc(Ω) |
if u∈W1,qloc(Ω) and if qq−1γ1≤q∗, where q∗ is the Sobolev conjugate exponent of q, and b1∈Lqq−1loc(Ω). On γ1 equivalently we require (if q<n) γ1≤q∗q−1q=nqn−qq−1q=n(q−1)n−q, which essentially corresponds to our assumption (3.8) below (the difference being the strict sign "<" for compactness reasons). We also observe that the summability condition b1∈Lqq−1loc(Ω) is satisfied if b1∈Ls1loc(Ω), with s1>nq−1, as in (3.10).
Similar computations apply to |b(x,u,ξ)φ|, again if q<n and with conjugate exponents q∗q∗−1 and q∗,
|b(x,u,Du)φ|≤Λ{|Du|r+|u|γ2+b2(x)}|φ|≤Λq∗−1q∗{|Du|r+|u|γ2+b2(x)}q∗q∗−1+Λq∗|φ|q∗∈L1loc(Ω) |
and we obtain b2∈Lq∗q∗−1loc(Ω) (compare with (3.10), where b2∈Ls2loc(Ω) with s2>np, since q∗q∗−1≤p∗p∗−1≤p∗p∗−p=np) and the conditions for r and γ2 expressed by rq∗q∗−1≤q and γ2q∗q∗−1≤q∗; i.e., for the first one,
r≤qq∗−1q∗=qnqn−q−1nqn−q=q+qn−1, |
which correspond to the more strict assumption (3.9), with r<p+pn−1, with the sign "<" and where q is replaced by p. Finally for γ2 we obtain γ2≤q∗−1, which again corresponds to our assumption (3.8) with the strict sign.
Therefore our assumptions for Theorem 3.2 are more strict than that ones considered in this section and they are consistent with a correct definition of weak solution to the elliptic equation (1.1).
Let ai:Ω×R×Rn→R, i=1,...,n, and b:Ω×R×Rn→R be Carathéodory functions, Ω be an open set in Rn, n≥2. Consider the nonlinear partial differential equation
n∑i=1∂∂xiai(x,u,Du)=b(x,u,Du). | (3.1) |
For the sake of simplicity we use the following notation: a(x,u,ξ)=(ai(x,u,ξ))i=1,...,n, for all i=1,…,n.
We assume the following properties:
● p−ellipticity condition at infinity:
there exist an exponent p>1 and a positive constant λ such that
⟨a(x,u,ξ),ξ⟩≥λ|ξ|p, | (3.2) |
for a.e. x∈Ω, for every u∈R and for all ξ∈Rn such that |ξ|≥1.
● q−growth condition:
there exist exponents q≥p, γ1≥0, s1>1, a positive constant Λ and a positive function b1∈Ls1loc(Ω) such that, for a.e. x∈Ω, for every u∈R and for all ξ∈Rn,
|a(x,u,ξ)|≤Λ{|ξ|q−1+|u|γ1+b1(x)}; | (3.3) |
● growth conditions for the right hand side b(x,u,ξ):
there exist further exponents r≥0, γ2≥0, s2>1 and a positive function b2∈Ls2loc(Ω) such that
|b(x,u,ξ)|≤Λ{|ξ|r+|u|γ2+b2(x)}, | (3.4) |
for a.e. x∈Ω, for every u∈R and for all ξ∈Rn.
Without loss of generality we can assume Λ≥1 and b1,b2≥1 a.e. in Ω. We recall the definition of weak solution to (3.1).
Definition 3.1. A function u∈W1,qloc(Ω) is a weak solution to (3.1) if
∫Ω{n∑i=1ai(x,u,Du)φxi+b(x,u,Du)φ}dx=0 | (3.5) |
for all φ∈W1,q(Ω), suppφ⋐Ω.
Our aim is to study the local boundedness of weak solutions to (3.1). Since this regularity property is trivially satisfied for functions in W1,qloc(Ω) with q>n, from now on we only consider the case q≤n; more precisely
1<p<n,p≤q≤n, | (3.6) |
since if q>n then weak solutions are Hölder continuous as an application of the Sobolev-Morrey embedding theorem, see Remark 3.3.
Other assumptions on the exponents are
{q<1+pqp<1+1n−1 | (3.7) |
0≤γ1<n(q−1)n−p,0≤γ2<n(p−1)+pn−p, | (3.8) |
0≤r<p+pn−1, | (3.9) |
s1>nq−1,s2>np. | (3.10) |
Under the conditions described above the following local boundedness result holds.
Theorem 3.2 (Boundedness result). Let u∈W1,qloc(Ω), 1<q≤n, be a weak solution to the elliptic equation (3.1). If (3.2)–(3.4) and (3.6)–(3.10) hold true, then u is locally bounded. Precisely, for every open set Ω′⋐Ω there exist constants R0,c>0 depending on the data n,p,q,r,γ1,γ2,s1,s2 and on the norm ‖u‖W1,q(Ω′) such that ‖u‖L∞(BR/2(x0))≤c for every R≤R0, with BR0(x0)⊆Ω′.
Remark 3.3. We already observed that if q>n then the weak solutions to (3.1) are locally Hölder continuous. Let us now discuss why in (3.6) we do not consider the case p=q=n. If p=q (≤n), the same computations in the proof of Theorem 3.2 work with the set of assumptions (3.8)–(3.10). They can be written, coherently with the previous ones, as
0≤γ1<p∗p−1p,0≤γ2<p∗−1 | (3.11) |
0≤r<p−pp∗, | (3.12) |
s1>p∗p(p∗−p)(p−1),s2>p∗p∗−p. | (3.13) |
Here p∗ denotes the Sobolev exponent appearing in the Sobolev embedding theorem for functions in W1,p(Ω) with Ω bounded open set in Rn; i.e.,
p∗:={npn−p if p<nany real number >n, if p=n. | (3.14) |
Following the computations in [40,Theorem 2.1] and [41,Chapter 6] it can be proved that the weak solutions to (3.1) are quasi-minima of the functional
F(u):=∫Ω(|Du|p+|u|τ+bpp−11+bp∗p∗−12)dx, | (3.15) |
with τ:=max{γ1pp−1,γ2p∗p∗−1}. It is known that if
τ<p∗andbpp−11+bp∗p∗−12∈L1+δ with δ>0 | (3.16) |
then the gradient of quasi-minima of the functional (3.15) satisfies a higher integrability property; i.e., they belong to W1,p+ϵ, for some ϵ>0.
Under our assumptions, (3.16) is satisfied; indeed, taking into account that we are considering p=q, by (3.10)
s1>np−1≥pp−1 |
and, by (3.13)
s2>p∗p∗−p≥p∗p∗−1. |
Analogously, by (3.11),
γ1pp−1<p∗,γ2p∗p∗−1<(p∗−1)p∗p∗−1=p∗. |
In particular, if p=q=n the quasi-minima of (3.15) are in W1,n+ϵloc(Ω) for some ϵ>0, therefore the weak solutions to (3.1) are Hölder continuous. We refer to [41] Chapter 6 for more details.
If p≥1 and d∈N, d≥2, we define
(pd)∗:={dpd−p if p<dany real number >d, if p=d. |
The Sobolev exponent appearing in the Sobolev embedding theorem for functions in W1,p(Ω), p≥1, with Ω bounded open set in Rn, is (pn)∗ and will be denoted, as usual, p∗.
Let t∈R, t>0. We define t∗ as follows:
1t∗:=min{1t+1n−1,1}. |
We have, if n≥3,
t∗={t(n−1)t+n−1if t>n−1n−21if 1≤t≤n−1n−2, |
and, if n=2, t∗=1 for every t.
We notice that, if n≥3,
((t∗)n−1)∗={tif t>n−1n−2n−1n−2if 1≤t≤n−1n−2 |
and, if n=2, for every t, ((t∗)n−1)∗ stands for any real number greater than 1.
Remark 4.1. Let us consider the exponents p,q satisfying (3.6) and (3.7) in Section 3. We notice that
1(pp−q+1)∗={1pp−q+1+1n−1if q>1+pn−11if q≤1+pn−1. | (4.1) |
Due to assumption (3.7), if n=2, then (pp−q+1)∗=1.
Moreover, if we denote t:=(pp−q+1)∗ then, if n≥3,
(tn−1)∗={pp−q+1if q>1+pn−1n−1n−2if q≤1+pn−1, | (4.2) |
if instead n=2 than (tn−1)∗ is any real number greater than 1.
Let p,q satisfy (3.6) and (3.7). It is easy to prove that
pp−q+1<q∗. | (4.3) |
In the following it will be useful to introduce the following notation:
ν:=1(pp−q+1)∗−1p, |
or, more explicitly,
ν={p−1pif q≤1+pn−11−qp+1n−1if q>1+pn−1. | (4.4) |
Remark 4.2. Assume 1<p≤q. Then easy computations give
ν>0⇔q<pnn−1,ν=0⇔q=pnn−1. | (4.5) |
To get the sharp bound for q, we use a result proved in [43], see also [3,4,30,58]. Here we denote Sσ(x0) the boundary of the ball Bσ(x0) in Rn.
Lemma 4.3. Let n∈N, n≥2. Consider Bσ(x0) ball in Rn and u∈L1(Bσ(x0)) and s>1. For any 0<ρ<σ<+∞, define
I(ρ,σ,u):=inf{∫Bσ(x0)|u||Dη|sdx:η∈C10(Bσ(x0)), 0≤η≤1, η=1 in Bρ(x0)}. |
Then for every δ∈]0,1],
I(ρ,σ,v)≤(σ−ρ)s−1+1δ(∫σρ(∫Sr(x0)|v|dHn−1)δdr)1δ. |
The following result is the Sobolev inequality on spheres.
Lemma 4.4. Let n∈N, n≥3, and γ∈[1,n−1[. Then there exists c depending on n and γ such that for every u∈W1,p(S1(x0),dHn−1)
(∫S1(x0)|u|(γn−1)∗dHn−1)1(γn−1)∗≤c(∫S1(x0)(|Du|γ+|u|γ)dHn−1)1γ. |
Lemma 4.5. Let n=2. Then there exists c such that for every u∈W1,1(S1(x0),dH1) and every r>1,
(∫S1(x0)|u|rdH1)1r≤c(∫S1(x0)(|Du|+|u|)dH1). |
Proof. By the one-dimensional Sobolev inequality
‖u‖L∞(S1(x0))≤c‖u‖W1,1(S1(x0)). |
Then, for every r>1,
(∫S1(x0)|u|rdHn−1)1r≤c‖u‖L∞(S1(x0))≤c‖u‖W1,1(S1(x0)). |
We conclude this section, by stating a classical result; see, e.g., [41]. that will be useful to prove Theorem 3.2.
Lemma 4.6. Let α>0 and (Jh) a sequence of real positive numbers, such that
Jh+1≤AλhJ1+αh, |
with A>0 and λ>1.
If J0≤A−1αλ−1α2, then Jh≤λ−hαJ0 and limh→∞Jh=0.
Under the assumptions in Section 3 we have the following Caccioppoli-type inequality.
Given a measurable function u:Ω→R, with Ω open set in Rn, and fixed x0∈Rn, k∈R and τ>0, we denote the super-level set of u as follows:
Ak,τ(x0):={x∈Bτ(x0):u(x)>k}; |
usually dropping the dependence on x0. We denote |Ak,τ| its Lebesgue measure.
Proposition 5.1 (Caccioppoli's inequality). Let u∈W1,qloc(Ω) be a weak solution to (3.1). If (3.6)–(3.10) hold true, then there exists a constant c>0, such that for any BR0(x0)⋐Ω, 0<ρ<R≤R0
∫Bρ|D(u−k)+|pdx≤C(n,p,q,R0)(R−ρ)−(pp−q+1−1+pp−q+1(pp−q+1)∗)××‖(u−k)+‖pp−q+1W1,p(BR)|Ak,R|pp−q+1ν+c‖(u−k)+‖pγ1q−1W1,p(BR)|Ak,R|1−pγ1p∗(q−1)+c‖(u−k)+‖pp−rW1,p(BR)|Ak,R|1−1p∗p−rp+c‖(u−k)+‖γ2+1W1,p(BR)|Ak,R|1−γ2+1p∗+c‖(u−k)+‖γ2W1,p(BR)|Ak,R|1−γ2p∗+ckγ2‖(u−k)+‖W1,p(BR)|Ak,R|1−1p∗+c(kpγ1q−1+kγ2)|Ak,R|+c‖(u−k)+‖W1,p(BR)|Ak,R|1−1s2−1p∗+c|Ak,R|1−ps1(q−1) | (5.1) |
with ν as in (4.4) and c is a constant depending on n,p,q,r,R0, the Ls1-norm of b1 and the Ls2-norm of b2 in BR0.
Proof. Without loss of generality we assume that the functions b1,b2 in (3.3) are a.e. greater than or equal to 1 in Ω. We split the proof into steps.
Step 1. Consider BR0(x0)⋐Ω, 0<R02≤ρ<R≤R0≤1.
We set
A(ρ,R):={η∈C∞0(BR(x0)):η=1in Bρ(x0), 0≤η≤1}. | (5.2) |
For every η∈A(ρ,R) and fixed k>1 we define the test function φk as follows
φk(x):=(u(x)−k)+[η(x)]μfor a.e. x∈BR0(x0), |
with
μ:=pp−q+1 | (5.3) |
that is greater than 1 because q>1.
Notice that φk∈W1,q0(BR0(x0)), suppφk⋐BR(x0).
Step 2. Let us consider the super-level sets:
Ak,R:={x∈BR(x0):u(x)>k}. |
In this step we prove that
∫Ak,ρ|Du|pdx≤c{∫Ak,R|Dη|pp−q+1(u−k)pp−q+1dx+∫Ak,R((u−k)pγ1q−1+(u−k)pp−r+(u−k)γ2+1+(u−k)γ2)dx+c∫Ak,R(kγ2(u−k)+b2(u−k)+kpγ1q−1+kγ2+bpq−11)dx} | (5.4) |
for some constant c independent of u and η.
Using φk as a test function in (3.5) we get
I1:=∫Ak,R⟨a(x,u,Du),Du⟩ημdx=−μ∫Ak,R⟨a(x,u,Du),Dη⟩ημ−1(u−k)dx−∫Ak,Rb(x,u,Du)(u−k)ημdx=:I2+I3. | (5.5) |
Now, we separately consider and estimate Ii, i=1,2,3.
ESTIMATE OF I3
Using (3.4) we obtain
I3≤Λ∫Ak,Rημ{|Du|r(u−k)+|u|γ2(u−k)+b2(u−k)}dx. |
We estimate the right-hand side using the Young inequality, with exponents pr and pp−r, and (3.2). There exists c, depending on λ, Λ, n, p, r, such that
Λ|Du|r(u−k)≤λ4|Du|p+c(u−k)pp−r≤14⟨a(x,u,Du),Du⟩+c(u−k)pp−ra.e. in {|Du|≥1}. | (5.6) |
and, recalling that b2≥1,
Λ|Du|r(u−k)≤Λ(u−k)≤Λb2(u−k)a.e. in {|Du|<1}. |
Therefore,
I3≤14∫Ak,R∩{|Du|≥1}⟨a(x,u,Du),Du⟩ημdx+c∫Ak,Rημ{(u−k)pp−r+|u|γ2(u−k)+b2(u−k)}dx. | (5.7) |
Collecting (5.5)–(5.7) we get
34∫Ak,R∩{|Du|≥1}⟨a(x,u,Du),Du⟩ημdx≤I2−∫Ak,R∩{|Du|≤1}⟨a(x,u,Du),Du⟩ημdx+c∫Ak,Rημ{(u−k)pp−r+|u|γ2(u−k)+b2(u−k)}dx. |
Using (3.2) and (3.3) we get
3λ4∫Ak,R∩{|Du|≥1}|Du|pημdx≤I2+2Λ∫Ak,R∩{|Du|≤1}(|u|γ2+b1)ημdx+c∫Ak,Rημ{(u−k)pp−r+|u|γ2(u−k)+b2(u−k)}dx. | (5.8) |
ESTIMATE OF I2. For a.e. x∈Ak,R∩{η≠0} we have
μ|⟨a(x,u,Du),Dη⟩|(u−k)ημ−1≤μΛ{|Du|q−1+|u|γ1+b1}|Dη|(u−k)ημ−1. | (5.9) |
For a.e. x∈{|Du|≥1}∩Ak,R∩{η≠0}, by q<p+1 and the Young inequality with exponents pq−1 and pp−q+1, and noting that μ−1=μq−1p, we get
μΛ|Du|q−1|Dη|(u−k)ημ−1≤λ4|Du|pημ+c(λ,Λ)μpp−q+1|Dη|pp−q+1(u−k)pp−q+1. | (5.10) |
On the other hand we have
μΛ|Du|q−1|Dη|(u−k)ημ−1≤μΛ|Dη|(u−k)ημ−1 | (5.11) |
a.e. in {|Du|<1}∩Ak,R∩{η≠0}.
Therefore,
I2≤λ4∫Ak,R∩{|Du|≥1}|Du|pημdx+c(λ,Λ)μpp−q+1∫Ak,R∩{|Du|≥1}|Dη|pp−q+1(u−k)pp−q+1dx+∫Ak,R|Dη|(u−k)ημ−1dx+c∫Ak,R|Dη|ημ−1{|u|γ1+b1}(u−k)dx. |
By (5.8) and the inequality above, we get
λ2∫Ak,R∩{|Du|≥1}|Du|pημdx≤c(λ,Λ,p,q)∫Ak,R|Dη|pp−q+1(u−k)pp−q+1dx+∫Ak,R|Dη|ημ−1(|u|γ1+b1)(u−k)dx+c∫Ak,Rημ((u−k)pp−r+|u|γ2(u−k)+|u|γ2+b2(u−k)+b1)dx. |
Taking into account that b1≥1
∫Ak,R|Du|pημdx=∫Ak,R∩{|Du|≥1}|Du|pημdx+∫Ak,R∩{|Du|<1}|Du|pημdx≤∫Ak,R∩{|Du|≥1}|Du|pημdx+∫Ak,Rb1ημdx, |
therefore
∫Ak,R(|Du|p−b1)ημdx≤∫Ak,R∩{|Du|≥1}|Du|pημdx |
and we obtain
∫Ak,ρ|Du|pdx≤c∫Ak,R|Dη|pp−q+1(u−k)pp−q+1dx+∫Ak,R|Dη|ημ−1(|u|γ1+b1)(u−k)dx+c∫Ak,Rημ((u−k)pp−r+|u|γ2(u−k)+|u|γ2+b2(u−k)+b1)dx. | (5.12) |
We have
∫Ak,R|Dη|ημ−1|u|γ1(u−k)dx≤c(γ1)∫Ak,R|Dη|ημ−1(u−k)γ1+1dx |
+c(γ1)∫Ak,R|Dη|ημ−1kγ1(u−k)dx. |
By Hölder inequality with exponents pq−1 and pp−q+1, we get
∫Ak,R|Dη|ημ−1(u−k)γ1+1dx=∫Ak,R|Dη|(u−k)ημ−1(u−k)γ1dx |
≤c∫Ak,R|Dη|pp−q+1(u−k)pp−q+1dx+c∫Ak,Rηp(μ−1)q−1(u−k)pγ1q−1dx. |
Analogously,
∫Ak,R|Dη|ημ−1kγ1(u−k)dx≤c∫Ak,R|Dη|pp−q+1(u−k)pp−q+1dx |
+c∫Ak,Rηp(μ−1)q−1kpγ1q−1dx |
and
∫Ak,R|Dη|ημ−1b1(u−k)dx≤c∫Ak,R|Dη|pp−q+1(u−k)pp−q+1dx |
+c∫Ak,Rηp(μ−1)q−1bpq−11dx, |
obtaining
∫Ak,ρ|Du|pdx≤c{∫Ak,R|Dη|pp−q+1(u−k)pp−q+1dx+∫Ak,R((u−k)pγ1q−1+kpγ1q−1+bpq−11)dx+c∫Ak,R((u−k)pp−r+|u|γ2(u−k)+|u|γ2+b2(u−k)+b1)dx.}. |
Therefore,
∫Ak,ρ|Du|pdx≤c{∫Ak,R|Dη|pp−q+1(u−k)pp−q+1dx+∫Ak,R((u−k)pγ1q−1+(u−k)pp−r+(u−k)γ2+1+(u−k)γ2)dx+c∫Ak,R(kγ2(u−k)+kγ2+b2(u−k)+b1+kpγ1q−1+bpq−11)dx.}. |
Since b1≥1 and q<p+1, then
b1+bpq−11≤2bpq−11, |
and we get (5.4).
Step 3. In this step we prove that
∫Bρ|D(u−k)+|pdx≤C(n,p,q,R0)(R−ρ)−(pp−q+1−1+pp−q+1(pp−q+1)∗)××‖(u−k)+‖pp−q+1W1,p(BR(x0))|Ak,R|pp−q+1ν+c∫Ak,R((u−k)pγ1q−1+(u−k)pp−r+(u−k)γ2+1+(u−k)γ2)dx+c∫Ak,R(kγ2(u−k)+b2(u−k)+kpγ1q−1+kγ2+bpq−11)dx. | (5.13) |
We obtain this estimate starting by (5.4).
Consider τ∈(ρ,R) and define the function
S1(0)∋y↦w(y):=(u−k)+(x0+τy) |
where
S1(0):={y∈Rn:|y|=1}. |
This function w is in W1,(pp−q+1)∗(S1,dHn−1), with
1(pp−q+1)∗=min{1pp−q+1+1n−1,1}. | (5.14) |
Let us consider the case
q>1+pn−1. |
By (4.1) in Remark 4.1, we get
1(pp−q+1)∗=1pp−q+1+1n−1. | (5.15) |
By (4.2) and the Sobolev embedding theorem, see Lemma 4.4, we get
(∫S1|w|pp−q+1dHn−1)p−q+1p≤c(n,p,q)(∫S1(|Dw|(pp−q+1)∗+|w|(pp−q+1)∗)dHn−1)1/(pp−q+1)∗. | (5.16) |
When
q≤1+pn−1, |
we distinguish among two cases: n≥3 and n=2. If n≥3, by using Hölder's inequality, we get
(∫S1|w|pp−q+1dHn−1)p−q+1p≤c(n,p,q)(∫S1|w|n−1n−2dHn−1)n−2n−1, |
by (4.2) and the Sobolev embedding theorem, see Lemma 4.4, we obtain the inequality (5.16).
If n=2, then (pp−q+1)∗=1, then we obtain the inequality (5.16) by applying Lemma 4.5 with r=pp−q+1.
Let A(ρ,R) be as in (5.2). We apply Lemma 4.3, with
BR(x0)∋y↦v(y):=(u−k)pp−q+1+(y), |
that is a function in L1(BR(x0)). Using (5.16) and recalling that R02≤ρ<R≤R0, reasoning as in [30], we get
infA(ρ,R)∫BR(x0)|Dη|pp−q+1(u−k)pp−q+1+dx≤C(n,p,q,R0)(R−ρ)−(pp−q+1−1+pp−q+1(pp−q+1)∗)××(∫Rρ∫Sτ(0)(|D(u−k)+(x0+y)|(pp−q+1)∗+|(u−k)+(x0+y)|(pp−q+1)∗)dHn−1(y)dτ)pp−q+1/(pp−q+1)∗. | (5.17) |
By coarea formula, inequality (5.17) implies
infA(ρ,R)∫BR(x0)|Dη|pp−q+1(u−k)pp−q+1+dx≤C(n,p,q,R0)(R−ρ)−(pp−q+1−1+pp−q+1(pp−q+1)∗)××‖(u−k)+‖pp−q+1W1,(pp−q+1)∗(BR(x0)∖Bρ(x0)) |
and, taking into account (3.7), Remark 4.1 and (4.5)
(pp−q+1)∗<p⇔1(pp−q+1)∗>1p⇔ν>0⇔qp<1+1n−1, |
by Hölder's inequality we get
infA(ρ,R)∫BR(x0)|Dη|pp−q+1(u−k)pp−q+1+dx≤C(n,p,q,R0)(R−ρ)−(pp−q+1−1+pp−q+1(pp−q+1)∗)××‖(u−k)+‖pp−q+1W1,p(BR(x0))|Ak,R|pp−q+1ν | (5.18) |
By (5.4) we get
∫Ak,ρ|Du|pdx≤C(n,p,q,R0)(R−ρ)−(pp−q+1−1+pp−q+1(pp−q+1)∗)××‖(u−k)+‖pp−q+1W1,p(BR(x0))|Ak,R|pp−q+1ν+c∫Ak,R((u−k)pγ1q−1+(u−k)pp−r+(u−k)γ2+1+(u−k)γ2)dx+c∫Ak,R(kγ2(u−k)+b2(u−k)+kpγ1q−1+kγ2+bpq−11)dx. |
Since
∫Bρ|D(u−k)+|pdx=∫Ak,ρ|D(u−k)+|pdx=∫Ak,ρ|Du|pdx |
we get (5.13).
Step 4. In this step we estimate the integrals at the right hand side of (5.13).
Consider
J1:=∫Ak,R((u−k)pγ1q−1+(u−k)pp−r+(u−k)γ2+1+(u−k)γ2)dx. |
ESTIMATE OF J1.
By assumptions (3.8) and (3.9),
max{pγ1q−1,γ2+1,pp−r}<p∗. |
Therefore, by using Hölder inequality with exponent p∗(q−1)pγ1 we get
∫Ak,R(u−k)pγ1q−1dx≤(∫Ak,R(u−k)p∗dx)pγ1p∗(q−1)|Ak,R|1−pγ1p∗(q−1); |
Hölder inequality with exponent p∗p−rp implies
∫Ak,R(u−k)pp−rdx≤(∫Ak,R(u−k)p∗dx)1p∗p−rp|Ak,R|1−1p∗p−rp. |
Moreover, by using Hölder inequality with exponent p∗γ2+1 we get
∫Ak,R(u−k)γ2+1dx≤(∫Ak,R(u−k)p∗dx)γ2+1p∗|Ak,R|1−γ2+1p∗; |
by using Hölder inequality with exponent p∗γ2 we get
∫Ak,R(u−k)γ2dx≤(∫Ak,R(u−k)p∗dx)γ2p∗|Ak,R|1−γ2p∗. |
Therefore, by using the Sobolev embedding theorem
J1≤‖(u−k)+‖pγ1q−1W1,p(BR)|Ak,R|1−pγ1p∗(q−1)+‖(u−k)+‖pp−rW1,p(BR)|Ak,R|1−1p∗p−rp+‖(u−k)+‖γ2+1W1,p(BR)|Ak,R|1−γ2+1p∗+‖(u−k)+‖γ2W1,p(BR)|Ak,R|1−γ2p∗. |
Let us consider now the following integral in (5.13):
J2:=∫Ak,R(kγ2(u−k)+b2(u−k)+kpγ1q−1+kγ2+bpq−11)dx. |
Trivially,
∫Ak,Rkγ2(u−k)dx≤kγ2‖(u−k)+‖1p∗Lp∗(Ak,R)|Ak,R|1−1p∗≤kγ2‖(u−k)+‖W1,p(Ak,R)|Ak,R|1−1p∗. |
By assumption b2∈Ls2, s2>np=p∗p∗−p. Since p∗p∗−p>p∗p∗−1, then s2s2−1<p∗. Therefore, by Hölder inequality
∫Ak,Rb2(u−k)dx≤‖b2‖Ls2(Ak,R)‖(u−k)+‖Ls2s2−1≤‖b2‖Ls2(BR)‖(u−k)+‖Lp∗(Ak,R)|Ak,R|1−1s2−1p∗, |
which implies
∫Ak,Rb2(u−k)dx≤‖b2‖Ls2(BR)‖(u−k)+‖W1,p(BR)|Ak,R|1−1s2−1p∗. |
Now, b1∈Ls1 with s1>pq−1; by using Hölder inequality with exponent s1(q−1)p we get
∫Ak,Rbpq−11dx≤(∫Ak,Rbs11dx)ps1(q−1)|Ak,R|1−ps1(q−1). |
We obtain
J2≤kγ2‖(u−k)+‖W1,p((BR))|Ak,R|1−1p∗+(kpγ1q−1+kγ2)|Ak,R|+‖b2‖Ls2(BR)‖(u−k)+‖W1,p(BR)|Ak,R|1−1s2−1p∗+‖b1‖pq−1Ls1(BR)|Ak,R|1−ps1(q−1). |
Step 5. By Steps 3, 4 we get
∫Br|D(u−k)+|pdx≤C(n,p,q,R0)(R−ρ)−(pp−q+1−1+pp−q+1(pp−q+1)∗)××‖(u−k)+‖pp−q+1W1,p(BR)|Ak,R|pp−q+1ν+c‖(u−k)+‖pγ1q−1W1,p(BR)|Ak,R|1−pγ1p∗(q−1)+c‖(u−k)+‖pp−rW1,p(BR)|Ak,R|1−1p∗p−rp+c‖(u−k)+‖γ2+1W1,p(BR)|Ak,R|1−γ2+1p∗+c‖(u−k)+‖γ2W1,p(BR)|Ak,R|1−γ2p∗+ckγ2‖(u−k)+‖W1,p(BR)|Ak,R|1−1p∗+c(kpγ1q−1+kγ2)|Ak,R|+c‖b2‖Ls2(BR)‖(u−k)+‖W1,p(BR)|Ak,R|1−1s2−1p∗+c‖b1‖pq−1Ls1(BR)|Ak,R|1−ps1(q−1) |
and the inequality (5.1) follows.
Let u∈W1,qloc(Ω), 1<q≤n, be weak solution to (3.1). Consider Ω′⋐Ω an open set.
I case q>p. Let BR0(x0)⊆Ω′.
For every k≥0
∫BR0(x0)(u−k)p+dx+∫BR0(x0)|D(u−k)+|pdx≤∫BR0(x0)(|u|−k)p+χ{x∈BR0(x0):|u|>k}(x)dx+∫BR0(x0)|Du|pχ{x∈BR0(x0):|u|>k}(x)dx≤∫BR0(x0)(|u|p+|Du|p)χ{x∈BR0(x0):|u|>k}(x)dx≤(∫BR0(x0)(|u|q+|Du|q)dx)p/q|{x∈BR0(x0):|u|>k}|1−p/q≤‖u‖pW1,q(BR0(x0))|BR0(x0)|1−p/q. | (6.1) |
In particular, chosen R0 such that
|BR0(x0)|≤‖u‖−pqq−pW1,q(Ω′) |
we get
‖(u−k)‖W1,p(BR0(x0))<1∀k≥0. | (6.2) |
II case q=p. By a well known result by Giaquinta and Giusti [40], the gradient of the weak solution satisfies a higher integrability property: its gradient is in Lp+ε(BR0(x0)), for some ε>0 sufficiently small. Moreover, u∈Lp∗(BR0(x0)); because p=q, we can repeat the above argument with q replaced by p+ε so obtaining (6.1). R0>0 depends on the norm ‖u‖W1,p+ε(BR0(x0)). Again, by the Giaquinta and Giusti result, the norm ‖u‖W1,p+ε(BR0(x0)) can be estimated in terms of the ‖u‖W1,p(Ω′) for BR0(x0)⊆Ω′⋐Ω.
Finally, we can summarize: in both cases, either if q>p or if q=p, we can choose R0 such that (6.2) holds with R0>0 depending on the norm ‖u‖W1,q(Ω′). We also assume R0<1 such that |BR0|<1, 0<R≤R0.
Define the decreasing sequences
ρh:=R2+R2h+1=R2(1+12h). |
Fixed a positive constant d≥2, to be chosen later, define the increasing sequence of positive real numbers (kh)
kh:=d(1−12h+1),h∈N. |
Define the decreasing sequence (Jh),
Jh:=‖(u−kh)+‖pW1,p(Bρh(x0)). |
Notice that
ρ0=R,limρ→+∞R2(1+12h)=R2, |
k0:=d2,limh→+∞kh=d. |
Moreover, by (6.2),
Jh≤J0=‖(u−d2)+‖pW1,p(BR(x0))<1. |
Let us introduce the following notation:
τ:=max{pp∗p−q+1ν+(pp−q+1−1+pp−q+1(pp−q+1)∗),p∗}, | (6.3) |
θ:=min{pp∗p−q+1ν,p∗−pγ1q−1,p∗−pp−r,p∗−γ2−1,p∗−p,p∗(1−1s2)−1,p∗(1−ps1(q−1))} | (6.4) |
and
σ:=min{1p−q+1+p∗p−q+1ν,p∗p−p∗s1(q−1),p∗p(1−1s2)}, | (6.5) |
where ν is defined in (4.4).
Proposition 6.1 (Estimate of Jh+1). Let u∈W1,qloc(Ω) be a weak solution to (3.1). Assume (3.2)–(3.4) with the exponents satisfying the inequalities listed in Section 3.1. Then for every h∈N
Jh+1≤c(2τ)hdθJσh, | (6.6) |
where c is a constant depending on n,p,q,r,R0, the Ls1-norm of b1 and the Ls2-norm of b2 in BR0.
We precede the proof with the following remark.
Remark 6.2. We remark that, by assumptions (3.6)–(3.10), then τ,θ>0 and σ>1. As far as these inequalities are concerned, we remark that
p∗>p; |
ν>0(see (4.5)); |
1p−q+1+p∗p−q+1ν>1⇔p∗ν>p−q |
that is satisfied, because p≤q
p∗>pp−r⇔r<p−pp∗⇔r<p+pn−1; |
p∗>pγ1q−1⇔γ1<p∗q−1p⇔γ1<n(q−1)n−p; |
γ2<p∗−1; |
p∗p−p∗s1(q−1)>1⇔ps1(q−1)<1−pp∗⇔s1>nq−1 |
that is the first assumption in (3.10); this assumption also implies
s1>pq−1>0 |
that is equivalent to
1−ps1(q−1)>pp∗>0. |
By the second assumption in (3.10),
s2>np⇔s2>p∗p∗−p⇔p∗p(1−1s2)>1. |
Proof of Proposition 6.1. By (5.1), used with k=kh+1, ρ=ρh+1, R=ρh, we have
∫Bρh+1|D(u−kh+1)+|pdx≤C(n,p,q,R0)(ρh−ρh+1)−(pp−q+1−1+pp−q+1(pp−q+1)∗)××‖(u−kh+1)+‖pp−q+1W1,p(Bρh)|Akh+1,ρh|pp−q+1ν+c‖(u−kh+1)+‖pγ1q−1W1,p(Bρh+1)|Akh+1,R|1−pγ1p∗(q−1)+c‖(u−kh+1)+‖pp−rW1,p(Bρh+1)|Akh+1,R|1−1p∗p−rp+c‖(u−kh+1)+‖γ2+1W1,p(Bρh+1)|Akh+1,R|1−γ2+1p∗+c‖(u−kh+1)+‖γ2W1,p(Bρh+1)|Akh+1,R|1−γ2p∗+ckγ2h+1‖(u−kh+1)+‖W1,p(Bρh+1)|Akh+1,R|1−1p∗+c(kpγ1q−1h+1+kγ2h+1)|Akh+1,R|+c‖(u−kh+1)+‖W1,p(Bρh+1)|Akh+1,R|1−1s2−1p∗+c|Akh+1,R|1−ps1(q−1). | (6.7) |
Let us write the estimate above as
∫Bρh+1|D(u−kh+1)+|pdx≤c(ρh−ρh+1)−(pp−q+1−1+pp−q+1(pp−q+1)∗)H1+c(H2+H3+H4+H5+H6+H7+H8+H9). | (6.8) |
To estimate the sum at the right-hand side it is useful to remark that, for all h,
kh+1−kh=d2h+2 | (6.9) |
and
kh+1−kh<u−khin Akh+1,ρh. |
Since
|Akh+1,ρh|≤∫Akh+1,ρh(u−khkh+1−kh)p∗dx≤‖(u−k)+‖p∗Lp∗(Bρh)1(kh+1−kh)p∗, |
by the Sobolev inequality we get
|Akh+1,ρh|≤c(n,p)Jp∗ph(kh+1−kh)p∗, |
that, together with (6.9), gives
|Akh+1,ρh|≤c(n,p)Jp∗ph(2hd)p∗. | (6.10) |
Moreover,
‖(u−kh+1)+‖pW1,p(Bρh(x0))=∫Akh+1,ρh(u−kh+1)pdx+∫Akh+1,ρh|D(u−kh+1)|pdx≤∫Akh,ρh(u−kh)pdx+∫Akh,ρh|D(u−kh)|pdx≤Jh. | (6.11) |
Inequalities (6.10) and (6.11) imply that
‖(u−kh+1)+‖W1,p(Bρh(x0))|Akh+1,R|−1p∗≤c(n,p)J1phJp∗p(−1p∗)h(kh+1−kh)p∗(−1p∗) |
therefore, by (6.9),
‖(u−kh+1)+‖W1,p(Bρh(x0))|Akh+1,R|−1p∗≤c(n,p)(2hd)−1. | (6.12) |
This estimate, together with (6.10), implies:
H2≤c(n,p,q,γ1)(2hd)−pγ1q−1|Akh+1,R|≤c(n,p,q,γ1)(2hd)p∗−pγ1q−1Jp∗ph, | (6.13) |
and, analogously,
H3≤c(n,p,r)(2hd)p∗−pp−rJp∗ph, | (6.14) |
H4≤c(n,p,γ2)(2hd)p∗−γ2−1Jp∗ph, | (6.15) |
H5≤c(n,p,γ2)(2hd)p∗−γ2Jp∗ph, | (6.16) |
H8≤c(n,p)(2hd)−1|Akh+1,R|1−1s2≤c(n,p,s2)(2hd)p∗(1−1s2)−1Jp∗p(1−1s2)h, | (6.17) |
H9≤c(n,p,q,s1)(2hd)p∗(1−ps1(q−1))Jp∗p−p∗s1(q−1)h. | (6.18) |
Moreover, taking into account that
kh+1=d(1−12h+2)≤d, |
H6≤c(n,p)dγ2(2hd)p∗−1Jp∗ph=c(n,p)2h(p∗−1)dp∗−γ2−1Jp∗ph | (6.19) |
H7≤c(2hp∗dp∗−pγ1q−1+2hp∗dp∗−γ2)Jp∗ph. | (6.20) |
Let us now estimate H1.
Inequalities (6.10) and (6.11) imply
H1:=‖(u−kh+1)+‖pp−q+1W1,p(Bρh(x0))|Akh+1,ρh|pp−q+1ν≤c(n,p,q)J1p−q+1h(Jp∗ph(kh+1−kh)p∗)pp−q+1ν |
that gives
Taking into account that for every
we conclude that
(6.21) |
Collecting (6.13)–(6.21), by (6.8) we get
(6.22) |
Let us now add to both sides of (6.22) the integral .
By Hölder inequality
Since
the Sobolev embedding theorem gives
(6.23) |
Taking into account (6.10), we obtain
therefore, the inequality (6.23) implies
(6.24) |
Inequalities (6.22) and (6.24) give
(6.25) |
where is a constant depending on , the -norm of and the -norm of in .
By taking in account the notation in (6.3)–(6.5), we get, by (6.25), the inequality (6.6).
We are now ready to prove our regularity result.
Proof of Theorem 3.2. By Proposition 6.1, for every ,
where is a constant depending on , the -norm of and the -norm of in and for every . Thus, the following inequality holds:
with
where , and are defined in (6.4), (6.3), (6.5). We recall that , see Remark 6.2.
To apply Lemma 4.6, we need
(6.26) |
Since
if we choose satisfying
(6.27) |
we get and we conclude that
To prove that is locally bounded from below, we proceed as follows. The function is a weak solution to
where
Notice that, by (3.2)–(3.4) the following properties hold:
● ellipticity condition at infinity:
for a.e. and for every ,
● growth condition:
for a.e. and every and
● growth condition for the right hand side :
To prove the analogue of Proposition 5.1 we now consider the test function where is a cut-off function. Let us consider the sub-level sets:
Then we obtain, in place of (5.5),
The proof goes on with no significant changes with respect the previous case, arriving to the conclusion that there exists such that we obtain that , and
Collecting the estimates from below and from above for , we conclude.
The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
The authors declare no conflict of interest.
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