Regularity of Weak Solutions of Elliptic and Parabolic Equations with Some Critical or Supercritical Potentials

TL;DR

This paper establishes Hölder continuity and differentiability of weak solutions to elliptic and parabolic equations with critical or supercritical potentials, extending classical regularity results beyond the De Giorgi-Nash-Moser theory.

Contribution

It proves regularity of solutions with potentials involving critical or supercritical singularities, which were previously outside the scope of classical theories.

Findings

Weak solutions are Hölder continuous under critical/supercritical potentials.

In some cases, weak solutions are differentiable.

Provides new regularity results for equations with singular potentials.

Abstract

We prove H\"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\Delta u-\frac{A}{|x|^{2+\beta}}u=0,\,\,(\beta\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01} \partial_{i} (a_{ij}(x) \partial_{j}u(x)) - \frac{A}{|x|^{2+\beta}} u(x) =0\quad (A>0,\quad\beta\geq 0), \end{equation} \begin{equation}\label{02} \partial_{i} (a_{ij}(x,t) \partial_{j}u(x,t)) - \frac{A}{|x|^{2+\beta}} u(x,t)-\partial_{t}u(x,t) =0\quad (A>0,\quad\beta\geq 0), \end{equation} with critical or supercritical 0-order term coefficients which are beyond De Giorgi-Nash-Moser's Theory. We also prove, in some special cases, weak solutions are even differentiable. Previously P. Baras and J. A. Goldstein \cite{Baras1984} treated the case when $A<0$, $(a_{ij})=I$ and $\beta=0$ for which they show that there does not exist any regular positive solution or singular positive solutions, depending on the size of $|A|$. When $A>0$, $\beta=0$ and $(a_{ij})=I$, P. D. Milman and Y. A. Semenov \cite{Milman2003}\cite{Milman2004} obtain bounds for the heat kernel.