Weighted-W^{1,p} estimates for weak solutions of degenerate and singular elliptic equations

Abstract

Global weighted L p -estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a weight in a Muckenhoupt class. Under a smallness condition on the mean oscillation of the coefficients with the weight and a Reifenberg flatness condition on the boundary of the domain, we establish a weighted gradient estimate for weak solutions of the equation. A class of degenerate coefficients satisfying the smallness condition is characterized. A counter example to demonstrate the necessity of the smallness condition on the coefficients is given. 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 in 1982.

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

“…As noted in [19,20,32,33], this space is different from the usual weighted BMO space, and is also different from the well-known John-Nirenberg BMO space. We also note that the smallness condition (1.10) is natural as it reduces to the regular smallness condition in BMO space that already used in known work [1,3,5,6,7,8,9,12,13,23,24,25,26,27,28,29,21,44]. Moreover, as demonstrated by a counterexample in [8], the smallness condition (1.10) is necessary.…”

“…Moreover, as demonstrated by a counterexample in [8], the smallness condition (1.10) is necessary. In particular, as it is shown in [8], the estimate (1.11) is even not valid when the coefficientsà is uniformly continuous, but degenerate. In this light and compared to [14], Theorem 1.3, once reduced to the linear case, gives the right conditions on the coefficients so that the W 1,p -regularity estimates hold.…”

“…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.…”

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

“…Moreover, we do not require the continuity of A in u. We also refer the readers to [2,4,5,6,7,8,11,12,18,19,20,24,25,26,27,17,36] for other papers in the same directions but only for linear equations with uniformly elliptic, bounded coefficients or for nonlinear equations in which A is independent on u. Indeed, it is well-known that the establishment of theory of Calderón-Zygmund estimates relies heavily on scaling invariant, see [36] for the geometric intuition.…”

“…Therefore, the Calderón-Zygmund estimates are usually available only for the PDEs which are invariant under this dilation. For example, see [2,4,5,6,7,8,11,12,18,19,20,24,25,26,27,17,36] for which linear equations and nonlinear equations where A is independent on u are studied, and those equations are invariant under the dilations u → u λ . However, as A depends on u, the equation (1.1) will be changed under the dilations u → u λ and this creates a serious issue.…”