Weighted W^1,p- estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable

2016/12/21 by Mengesha, Tadele, Phan, Tuoc
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1612.07371

Abstract

This paper studies the Sobolev regularity of weak solution of degenerate elliptic equations in divergence form div[A(X) ∇ u] = div[F(X)], where X = (x,y) ∈ ℝⁿ × ℝ . The coefficient matrix A(X) is a symmetric, measurable (n+1) × (n+1) matrix, and it could be degenerate or singular in the one dimensional y-variable as a weight function in the Muckenhoupt class A₂ of weights. Our results give weighted Sobolev regularity estimates of Calderón-Zygmund type for weak solutions of this class of singular, degenerate equations. As an application of these estimates, we establish global Sobolev regularity estimates for solutions of the spectral fractional elliptic equation with measurable coefficients. This result can be considered as the Sobolev counterpart of the recently established Schauder regularity theory of fractional elliptic equations.

Related

• Weighted Calderón-Zygmund estimates for weak solutions of quasi-linear degenerate elliptic equations
• Weighted-W^1,p estimates for weak solutions of degenerate and singular elliptic equations
• Regularity theory for parabolic equations with singular degenerate coefficients
• Matrix Weights and Regularity for Degenerate Elliptic Equations
• Sobolev estimates for parabolic and elliptic equations in divergence form with degenerate coefficients