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

Abstract

This paper establishes Calderón-Zygmund type regularity estimates for solutions of the conormal derivative problem for a class of linear elliptic systems in divergence-form with singular, degenerate coefficients in bounded domains. In our class of equations, the principal terms are fourth order tensors of measurable functions that behave as some weight function in the Muckenhoupt class of A 2-weights. Regularity estimates for gradient of weak solutions in weighted Lebesgue spaces are established under some natural smallness conditions on the mean oscillation of coefficients. The results obtained recover known results when the coefficients are uniformly elliptic. These results can be considered as the Sobolev counterparts of the classical Hölder's regularity estimates established by B. Fabes, C.

Elliptic Equations With Degenerate weights

“…Their condition allowed to include weights like |x| ±ε Id for small ε > 0. The case of systems has been covered by the same authors in [7]. Our condition on the weight A differs somehow from the previous ones.…”

Global Maximal Regularity for Equations with Degenerate Weights

“…They yield |F | q µ ∈ L 1 loc ⇒ |∇u| q µ ∈ L 1 loc for all q ∈ (1, ∞), including the case µ(x) = |x| ±ε Id for small ε > 0. The global result is obtained in [Pha20] and the local result for the case of systems is proved in [CMP19]. In the recent paper [BDGN21], the authors prove a new type of gradient estimates with the implication that (|F |ω) q ∈ L 1 loc ⇒ (|∇u|ω) q ∈ L 1 loc for all q ∈ (1, ∞), assuming (1.10) and the smallness condition for the BMO norm of log A as follows:…”

“…In [CMP18], they introduced the quantity 1 B |A(x) − A B | 2 µ −1 (x) dx measure the oscillations of A, where µ(x) := |A(x)| and the supremum is taken over all balls. In addition to the smallness conditions Cao, Mengesha and Phan assume that µ := |A| ∈ A 2 .In[CMP19] they use the simpler quantity|A| BMOµ := sup B 1 µ(B) B |A(x) − A B | dx. (4.2)Note that by Hölder's inequality|A| BMOµ ≤ |A| BMO 2 µ .…”