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