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

Abstract

We prove Hölder continuity of weak solutions of the uniformly elliptic and parabolic equations %$\Delta u-\frac{A}{|x{|}^{2+\beta }}u=0,(\beta \ge 0)$, and variable second order term coefficients case A<0$,$(a_{ij})=I$and$β=0$forwhichtheyshowthattheredoesnotexistanyregularpositivesolutionorsingularpositivesolutions,dependingonthesizeof$|A|$.When$A>0$,$β=0$and$(a_{ij})=I$, P. D. Milman and Y. A. Semenov \cite{Milman2003}\cite{Milman2004} obtain bounds for the heat kernel.

arXiv comment32 pages