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

Abstract

This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet boundary conditions over bounded spatial domains. Our main focus is on the case that the vector coefficients $\mathbb{A}$ are discontinuous and singular in $(x,t)$-variables, and dependent on the solution $u$. Global and interior weighted $W^{1,p}(\Omega ,\omega )$-regularity estimates are established for weak solutions of these equations, where $\omega$ is a weight function in some Muckenhoupt class of weights. The results obtained are even new for linear equations, and for $\omega =1$, because of the singularity of the coefficients in $(x,t)$-variables

arXiv admin note: text overlap with arXiv:1703.02706. text overlap with arXiv:1702.08622

References in corpus (1)

• Weighted Calderón-Zygmund estimates for weak solutions of quasi-linear degenerate elliptic equations