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Register a new userA new short proof of regularity for local weak solutions for a certain class of singular parabolic equations
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We shall establish the interior Holder continuity for locally bounded weak solutions to a class of parabolic singular equations whose prototypes are begin{equation} u_t= abla cdot bigg( | abla u|^{p-2} abla u bigg), quad text{ for } quad 1<p<2, end{equation} and begin{equation} u_{t}- abla cdot ( u^{m-1} | abla u |^{p-2} abla u ) =0 , quad text{for} quad m+p>3-frac{p}{N}, end{equation} via a new and simplified proof using recent techniques on expansion of positivity and $L^{1}$-Harnack estimates.
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This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - mbox{div}[mathbb{A}(x,t,u, abla u)]= mbox{div}[{mathbf F}]$ 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
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