Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations

Abstract

We study a series of regularity results for solutions to a degenerate or singular fully nonlinear integro-differential equation of the form $$-(\ddot{I}{ƒ}_1(|Du|)+a(x)\ddot{I}{ƒ}_2(|Du|)){\mathcal{I}}_{\ddot{I}}„(u,x)=f(x).$$ In the degenerate case, we establish borderline regularity, provided the inverse of the degeneracy law $\ddot{I}{ƒ}_2$ is Dini-continuous. In addition, we show Schauder-type higher regularity at local extremum points for a specific non-local degenerate equation. In the singular case, we establish Hölder continuity of the gradient for solutions to a general non-local equation. It is noteworthy that these results are new even in the case $a(x)\equiv 0$. Finally, as a byproduct of the borderline regularity analysis, we demonstrate how our methods can be applied to study of the corresponding regularity for a class of degenerate non-local normalized $p$-Laplacian equations.

28 pages, Referees'suggestions incorporated, To appear in Communications in Contemporary Mathematics