Can I read H\"older Regularity for Nonlocal Double Phase Equations on EtoBox?

H\"older Regularity for Nonlocal Double Phase Equations by De Filippis, Cristiana; Palatucci, Giampiero is a scholarly article available to read on EtoBox.

What is H\"older Regularity for Nonlocal Double Phase Equations about?

We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient $a=a(\cdot,\cdot)$. The model case is driven by the following nonlocal double phase operator, $$ \int \!\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\, {\rm d}y + \int \!a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\, {\rm d}y, $$ where $q\geq p$ and $a(\cdot,\cdot)\geqq 0$. Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are H\"older continuous, whereas similar sharp results for the classical local case do require $a$ to be H\"older continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.

Author

De Filippis, Cristiana; Palatucci, Giampiero

Published

2019

Language

EN