On optimal estimates for -Laplace type equations☆
Introduction
In the present work, we study the following -Laplace elliptic equation where is a bounded domain, and variable exponents satisfy Throughout this paper, the coefficient admits the uniform elliptic condition for any with . Here, we assume without loss of generality. The functions , and are supposed to satisfy the continuity conditions: for all with positive constants . The weak solutions of Eq. (1.1) are understood in the weak sense as below.
Definition 1.1 A function is a local weak solution to (1.1) in provided that for any function .
Before exhibiting our main result, we would like to demonstrate the motivation of this work. The studies of optimal regularity for -Laplace equations have attracted extensive attentions during the last decades. We begin with introducing the very known -conjecture for the following elliptic equation with . The conjecture tells that the solutions to Eq. (1.8) are locally of class provided and . Note that solves and . This fact as supportive evidence for the above conjecture indicates the optimal Hölder exponent of the gradients cannot be greater than . Recently, Araújo et al. proved the -conjecture in the case that is a plane [1], and investigated the optimal -exponent in the higher dimension [2]. Besides, under the assumption that in (1.8) admits -integrability with , 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 When the field in Eq. (1.9) admits the -laplace type structure and Hölder continuity conditions, the interior sharp exponent for was determined in [3] provided that the nonhomogeneous term satisfies a certain -integrability. Based on the aforementioned results, Haque [16] recently considered Eq. (1.1) with and being constants, and studied the global optimal Hölder exponent of when Eq. (1.1) is subject to Dirichlet or Neumann boundary condition.
Except for the sharp exponent of , 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 with and showed that with . Balci et al. [4] transferred the local interior Besov and Triebel–Lizorkin regularity up to first order derivatives from the force term to the flux for the dimension . In [11], [19], the authors studied Eq. (1.9) and gave the essentially optimal condition for -regularity, specifically, the modulus of continuity for is obtained under the hypotheses that is Dini-continuous and with . In addition, we can find the corresponding optimal regularity results for the parabolic -Laplace equations (see the survey in [4], [8]).
The aim of the present work is to establish the optimal estimate for the -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) , 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 only depending on and such that . Hence for any weak solution of (1.1), this fact allows us to operate in a pointwise way.
To simplify our statements let us invoke some notations. For any and , we denote , by Let the solutions of the standard -Laplacian equation be locally -continuous with the maximal exponent . Although the -conjecture has not been completely solved, it is known from [9], [20], [29] that only depends on and . Then is taken as Moreover, we also need to define the quantity 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 is a local weak solution of (1.1) under the assumptions (1.2)–(1.6). Let any and be given as with any . Then we have . Specifically, there holds for some only depending on and .
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 and , for any , the weak solutions of Eq. (1.1) belong to with
The properties of generalized Lebesgue spaces associated with the variable exponent equations are necessary tools to investigate the solution of (1.1). For satisfying in , we denote by the variable exponent Lebesgue space equipped with the Luxemburg norm It is well known that is a separable and reflexive Banach space. We write if with any . When is a constant, variable exponent Lebesgue spaces coincide with classical Lebesgue spaces. For any , the modules are defined as
The result given in the next lemma can help us to handle the nonhomogeneous term by using the integral in the modules form.
Lemma 1.1 See [15], [18] For any , we have
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 -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 for any 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 () 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 () 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 to denote the euclidean distance between and the boundary of the domain ; a function with an integer and if for any and each ; is a open ball with radius centered at , and we simplify the symbol by writing when the center is the origin. In the following sections, if a constant only depends on and , then we call this constant is universal.
Section snippets
Regularity estimates for large radii
In this section, we focus on the point satisfying , and then aim at obtaining the estimates (1.16).
For the sake of clarity, we manage the solution on the unit ball centered at the origin, and also take the discussed point in (1.16) to be the origin. The desired result is derived by utilizing the oscillation of a proper corrector and estimating with . Hence, let us turn to the reference equation with positive
Regularity estimates for small radii
In this section, we first provide an optimal Hölder exponent result for the following equation where with a variable exponent satisfying (1.3), the fields and are assumed to admit for any and with positive constants . The nondecreasing module functions in (3.4) satisfy and
References (31)
- et al.
A proof of the conjecture in the plane
Adv. Math.
(2017) - et al.
Higher order Calderón–Zygmund estimates for the -Laplace equation
J. Differential Equations
(2020) - et al.
Hölder continuity of the gradient of -harmonic mappings
C. R. Acad. Sci. Paris I
(1999) Geometric regularity estimates for nonlinear evolution models
Nonlinear Anal.
(2019)Local regularity of weak solutions of degenerate elliptic equations
Nonlinear Anal.
(1983)Global regularity for variable exponent elliptic equations in divergence form
J. Differential Equations
(2007)- et al.
Existence of solutions for -Laplacian Dirichlet problem
Nonlinear Anal.
(2003) - et al.
A class of De Giorgi type and Hölder continuity
Nonlinear Anal.
(1999) - et al.
On the spaces and
J. Math. Anal. Appl.
(2001) A note on the optimal boundary regularity for the planar generalized -Poisson equation
Nonlinear Anal.
(2018)
Overview of differential equations with non-standard growth
Nonlinear Anal.
Universal potential estimates
J. Funct. Anal.
Nonlinear elliptic equations with variable exponent: old and new
Nonlinear Anal.
Regularity for a more general class of quasilinear elliptic equations
J. Differential Equations
Global -estimates and Hölder continuity of weak solutions to elliptic equations with the general nonstandard growth conditions
Nonlinear Anal.
Cited by (10)
A second-order Sobolev regularity for p(x)-Laplace equations
2023, Journal of Mathematical Analysis and ApplicationsLocal Hölder regularity for the general non-homogeneous parabolic equations
2023, Journal of Mathematical Analysis and ApplicationsThe boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth
2022, Journal of Differential EquationsLocal regularity for nonlocal equations with variable exponents
2023, Mathematische NachrichtenON THE SOLVABILITY OF VARIABLE EXPONENT DIFFERENTIAL INCLUSION SYSTEMS WITH MULTIVALUED CONVECTION TERM
2023, Rocky Mountain Journal of Mathematics