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

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

No reviews yet.

No comments yet.