Higher Holder regularity for nonlocal equations with irregular kernel

We study the higher Holder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore, in some sense our result can be considered to be of purely nonlocal type, following the trend of various such purely nonlocal phenomena observed in recent years. Our approach can be summarized as follows. First, we use certain test functions that involve discrete fractional derivatives in order to obtain higher Holder regularity for homogeneous equations driven by a locally translation invariant kernel, while the global behaviour of the kernel is allowed to be more general. This enables us to deduce the desired regularity in the general case by an approximation argument.

Automated summary

The study focuses on the higher Holder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels satisfying a mild continuity assumption. The obtained regularity is better than expected compared to results for local elliptic equations with continuous coefficients, showcasing a purely nonlocal trend. The approach involves using test functions with discrete fractional derivatives to achieve higher Holder regularity for homogeneous equations, ultimately deducing desired regularity through an approximation argument.