Elsevier

Nonlinear Analysis

Volume 200, November 2020, 112030
Nonlinear Analysis

On optimal C1,α estimates for p(x)-Laplace type equations

https://doi.org/10.1016/j.na.2020.112030Get rights and content

Abstract

In this paper, we investigate the optimal C1,α estimates for the elliptic p()-Laplace equation: div(a(x)|u|p(x)2u)=divh(x)+f(x)inΩwith fLq()(Ω) and a,hCσ(Ω¯). Based on a certain geometric oscillation estimate, the scaling arguments and appropriate localization technique as well as the careful analysis on the variable exponents, we exhibit how the optimal Hölder exponent of u is influenced by p(), q() and σ. This work can be regarded as a natural follow up to the paper by Araújo and Zhang (in press).

Introduction

In the present work, we study the following p(x)-Laplace elliptic equation diva(x)|u|p(x)2u=divh(x)+f(x)inΩ,where ΩRn is a bounded domain, fLq()(Ω) and variable exponents p(),q()C(Ω¯) satisfy 1<pinfΩ¯p(x)supΩ¯p(x)p+<,n<qinfΩ¯q(x)supΩ¯q(x)q+. Throughout this paper, the coefficient a() admits the uniform elliptic condition λa(x)Λfor any xΩ with 0<λ<Λ<. Here, we assume Λλ without loss of generality. The functions a(), p() and h() are supposed to satisfy the continuity conditions: |a(x)a(y)|+|p(x)p(y)|C1|xy|σ1, |h(x)h(y)|C2|xy|σ2for all x,yΩ¯ with positive constants C1,C2>0. The weak solutions of Eq. (1.1) are understood in the weak sense as below.

Definition 1.1

A function uWloc1,p()(Ω) is a local weak solution to (1.1) in Ω provided that Ωa(x)|u|p(x)2uφdx=Ωh(x)φdxΩf(x)φdxfor any function φC0Ω.

Before exhibiting our main result, we would like to demonstrate the motivation of this work. The studies of optimal regularity for p-Laplace equations have attracted extensive attentions during the last decades. We begin with introducing the very known Cp-conjecture for the following elliptic equation Δpudiv|u|p2u=finΩwith p>1. The conjecture tells that the solutions to Eq. (1.8) are locally of class C1,1p1 provided p>2 and fL(Ω). Note that v(x)=p1p|x|pp1 solves Δpv=n and vC1,1p1. This fact as supportive evidence for the above conjecture indicates the optimal Hölder exponent of the gradients cannot be greater than 1p1. Recently, Araújo et al. proved the Cp-conjecture in the case that Ω is a plane [1], and investigated the optimal C1,α-exponent in the higher dimension n3 [2]. Besides, under the assumption that f in (1.8) admits Lq-integrability with q(2,), the sharp Hölder exponent for the gradient was obtained by Lindgren and Lindqvist in a planar setting [21].

Let us turn to a quasilinear equation in the form of div(a(x,u))=f(x)inΩ.When the field a in Eq. (1.9) admits the p-laplace type structure and Hölder continuity conditions, the interior sharp exponent for C1,α was determined in [3] provided that the nonhomogeneous term f satisfies a certain Lq-integrability. Based on the aforementioned results, Haque [16] recently considered Eq. (1.1) with p() and q() being constants, and studied the global optimal Hölder exponent of u when Eq. (1.1) is subject to Dirichlet or Neumann boundary condition.

Except for the sharp exponent of C1,α, there are also some known literature concerned with other types of optimal regularity results. For the sharp Calderón–Zygmund estimate in Eq. (1.8), Brasco and Santambrogio [5] assumed fWlocs,p(Ω) with s(p1p,1] and showed that uWlocσ,p(Ω) with σ(0,2p). Balci et al. [4] transferred the local interior Besov and Triebel–Lizorkin regularity up to first order derivatives from the force term f to the flux |u|p2u for the dimension n=2. In [11], [19], the authors studied Eq. (1.9) and gave the essentially optimal condition for C1,0-regularity, specifically, the modulus of continuity for u is obtained under the hypotheses that a(,ξ) is Dini-continuous and fL(n,1p1) with p>21n. In addition, we can find the corresponding optimal regularity results for the parabolic p-Laplace equations (see the survey in [4], [8]).

The aim of the present work is to establish the optimal C1,α estimate for the p()-Laplacian problems which arise in the mathematical modeling of electro-rheological fluids developed by Růžička and Rajagopal [24], [25]. The solvability for the elliptic equations with nonstandard growth has been extensively investigated in [13], [17], [26], [27], [28] and references therein. For regularity results on the variable exponent equations, we refer the readers to [7], [12], [14], [23], [30], [31]. Here we only emphasized the Hölder continuity results which serve as an essential ingredient in our proof. When the right-hand side of (1.1) |h|=f0, the local Hölder continuity of the gradient of solutions has been established by Coscia and Mingione in [7]. Furthermore, under the assumptions (1.2)–(1.6), it can be deduced from [12], [31] that there is a constant β(0,1) only depending on p+,p,q+,q,σ1,σ2,n and Ω such that uCloc1,β(Ω). Hence for any weak solution u of (1.1), this fact allows us to operate u in a pointwise way.

To simplify our statements let us invoke some notations. For any gC(Ω¯) and VΩ, we denote gV, gV+ by gVinfV¯g(x),gV+supV¯g(x).Let the solutions v of the standard p-Laplacian equation Δpv=0inΩbe locally C1,αp-continuous with the maximal exponent αp. Although the Cp-conjecture has not been completely solved, it is known from [9], [20], [29] that αp only depends on p and n. Then αp(),Vm is taken as αp,Vminfp[pV,pV+]αp.Moreover, we also need to define the quantity σp(),q(),Vminσ1,σ2,1nqVmin1,1pV+1 if n<qV<,minσ1,σ2min1,1pV+1 if qV=.With these preparations, we are ready to present our main result which can be regarded as an extension of [3, Theorem 1.1] to the more general equations with variable exponent growth.

Theorem 1

Assume that uWloc1,p()(Ω) is a local weak solution of (1.1) under the assumptions (1.2)(1.6). Let any VΩ and α be given as α=min{αp(),Vmε,σp(),q(),V}with any ε(0,αp,Vm). Then we have uC1,α(V). Specifically, there holds supx,yV,xy|u(x)u(y)||xy|αCfor some C>0 only depending on σ1,σ2,α,n,Ω,pV,pV+,qV,qV+,Λ,λ,uW1,p(),C1,C2,fLq() and dist(V,Ω).

Based on the result given in [1], we can also establish the optimal exponent in a two-dimensional setting.

Theorem 2

Under conditions (1.2)(1.6) with 2<q(),p()2 and n=2, for any VΩ, the weak solutions of Eq. (1.1) belong to C1,ϱ(V) with ϱ=ϱp(),q(),Vminσ1,σ2,1nqV1pV+1 if n<qV<,minσ1,σ21pV+1 if qV=.

The properties of generalized Lebesgue spaces associated with the variable exponent equations are necessary tools to investigate the solution of (1.1). For p()CΩ¯ satisfying p(x)>1 in Ω¯, we denote by Lp()Ω the variable exponent Lebesgue space Lp()Ω=u|u:ΩR is measurable and Ω|u(x)|p(x)dx<equipped with the Luxemburg norm uLp()(Ω)=infλ>0:Ω|u(x)λ|p(x)dx1.It is well known that Lp()Ω is a separable and reflexive Banach space. We write uLlocp()(Ω) if uLp()(V) with any VΩ. When p() is a constant, variable exponent Lebesgue spaces coincide with classical Lebesgue spaces. For any uLp()(Ω), the modules are defined as ρ(u)Ω|u(x)|p(x)dx.

The result given in the next lemma can help us to handle the nonhomogeneous term f by using the integral in the modules form.

Lemma 1.1 See [15], [18]

For any uLq()(Ω), we have minuLq()(Ω)q,uLq()(Ω)q+ρ(u)maxuLq()(Ω)q,uLq()(Ω)q+.

By far, we only briefly introduce the basic information concerning the generalized Lebesgue spaces, which is needed in our forthcoming proofs. For more details on generalized Lebesgue spaces and Lebesgue–Sobolev spaces, we refer the readers to [10], [15], [18], [22].

Organization of the paper. This paper investigates p(x)-Laplace type equations and provides the corresponding results to those given in [2], [3]. The methods used here are derived from a series works of [2], [3], [16]. Hence, the core estimate of this work is supBρx0|u(x)ux0ux0xx0|Cρ1+αfor any ρ>0 small enough. To this end, we not only need to follow the arguments developed in [3], but also to carefully perform localization estimates for solving the difficulties caused by the anisotropy of the variable exponent. In Section 2, we first focus on the large radii (|u|r) and use the well-used arguments to get the oscillation estimate (1.16). Section 3 is devoted to dealing with the other case, namely, the small radii (r|u|) is considered and the desired estimate can be obtained by utilizing linear growth of the coefficient.

Notations. The symbols used in this paper are standard, such as, we use dist(V,Ω) to denote the euclidean distance between V and the boundary of the domain Ω; a function uClock,σ(Ω) with an integer k0 and σ(0,1) if DβuCσ(V) for any VΩ and each |β|k; Br(x) is a open ball with radius r>0 centered at x, and we simplify the symbol by writing Br=Br(0) when the center is the origin. In the following sections, if a constant only depends on Ω,V,n,σ1,σ2,pV,pV+,qV,qV+ and α, then we call this constant is universal.

Section snippets

Regularity estimates for large radii

In this section, we focus on the point x0 satisfying |ux0|ρα, and then aim at obtaining the estimates (1.16).

For the sake of clarity, we manage the solution u on the unit ball B1 centered at the origin, and also take the discussed point x0 in (1.16) to be the origin. The desired result is derived by utilizing the oscillation of a proper corrector v and estimating supB12|uv| with |u(0)v(0)|. Hence, let us turn to the reference equation divā|v|p̄2v=0 in B34,v(0)=0with positive

Regularity estimates for small radii

In this section, we first provide an optimal Hölder exponent result for the following equation divā(x,v)=divh(x)+f(x)inΩ,where fLq()(Ω) with a variable exponent q()C(Ω¯) satisfying (1.3), the fields ā and h are assumed to admit |ā(x,ξ)|+|ξā(x,ξ)||ξ|Λ̄|ξ|,λ̄|ξ2|2ξā(x,ξ1)ξ2,ξ2,|ā(x,ξ)ā(y,ξ)|ω1(|xy|)|ξ|,|h(x)h(y)|ω2(|xy|) for any x,yΩ and ξ1,ξ1,ξRn with positive constants λ̄,Λ̄>0. The nondecreasing module functions ωi (i=1,2) in (3.4) satisfy ωi(0)=0 and ωiCσi([0,)),i=1

References (31)

Cited by (10)

View all citing articles on Scopus

Supported by the National Natural Science Foundation of China (11671111, 11571020, 11671021).

View full text