Weighted Calderón-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

Abstract

This paper studies the Sobolev regularity estimates for weak solutions of a class of degenerate, and singular quasi-linear elliptic problems of the form div [A(x, u, ∇u)] = div[F] with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients A could be be singular, and degenerate or both in x in the sense that they behave like some weight function µ, which is in the A 2 class of Muckenhoupt weights. Global and interior weighted W 1,p (Ω, ω)-regularity estimates are established for weak solutions of these equations with some other weight function ω. The results obtained are even new for the case µ = 1 because of the dependence on the solution u of A. In case of linear equations, our W 1,p -regularity estimates can be viewed as the Sobolev's counterpart of the Hölder's regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.with p > 2, and some other weight function ω in some class of Muckenhoupt weights, whose definitions will be given later.The purpose of this paper is twofold. On one hand, it is the continuation of the developments of the recent work [2,17,36,37] on the theory of Sobolev regularity theory for weak solutions of quasi-linear elliptic equations in which the coefficients A are dependent on the solution u. See, for instance [1,3,5,6,7,8,9,12,13,23,24,25,26,27,28,29,21,44] for other work in the same directions but only for linear equations or for equations in which A is independent on u. On the other hand, this work includes the case that A could be singular or degenerate as a weight in some Munkenhoupt class of weights as considered in many papers such as [15,16,22,30,31,35,38,41,42] in which only Schauder's regularity of weak solutions are investigated. Moreover, even for the uniformly elliptic case, the results in this paper also improve those in [2,17,36,37] since they do not require a-priori boundedness of weak solutions of (1.1). This work also extends the recent work [8] to the nonlinear case and two weighted estimates. Results of this paper can be considered as the Sobolev's regularity counterpart of the Schauder's one established in [15,16,22,30,31,35,38,41,42] for singular, degenerate equations.

Gradient estimates for weak solutions of linear elliptic systems with singular-degenerate coefficients

“…Recently, in [4], we have obtained estimates in Lebesgue spaces for gradient of solutions to zero Dirichlet boundary value problems for linear degenerate equations of type (1.1) with general µ ∈ A 2 . Two weighted Calderón-Zygmund type regularity estimates for quasilinear elliptic equations with prescribed singular-degenerate coefficients and non-homogeneous Dirichlet boundary conditions are also established in [24]. We aim to extend the above mentioned results in [4,24] and establish the corresponding results for weak solutions of the conormal derivative problem for linear systems (1.1) by giving the right and optimal conditions on the coefficient tensor.…”

“…Two weighted Calderón-Zygmund type regularity estimates for quasilinear elliptic equations with prescribed singular-degenerate coefficients and non-homogeneous Dirichlet boundary conditions are also established in [24]. We aim to extend the above mentioned results in [4,24] and establish the corresponding results for weak solutions of the conormal derivative problem for linear systems (1.1) by giving the right and optimal conditions on the coefficient tensor. As already indicated, the regularity estimate we obtain requires that coefficient matrix must not oscillate too much.…”

“…See, for examples [4,Lemma 5.10] or [24,Proposition 4.9] for the details on the proof of this claim. Using this claim, and by induction, we then infer that for any k ∈ N, Observe that, when finite, S 2/p is comparable to the L p/2 -norm of M µ (χ B + 2 |∇u M |) in B + 1 .…”

Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations

“…In this paper, we overcome the inhomogeneity in (1.1) by adapting the perturbation technique with double-scaling parameter method introduced in [16]. See also [29,31,30] for the implementation of the method. We essentially enlarge and consider the following class of equations with scaling parameter…”

On Limits at Infinity of Weighted Sobolev Functions

“…Weighted Sobolev spaces, especially with Muckenhoupt weights, have played a crucial role in PDEs and the study of variational problems, starting from [FKS82]. They are still actively employed in these topics; see [Cav20,Pha20,Chu92]. Weighted function spaces have been further studied by many authors in regard to their intrinsic properties, such as regularity and the existence of traces; see [Ryc01, TS20, BBS20, Tyu13, Tyu14b, Tyu14a].…”