Abstract
We study the higher Hölder 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 Hölder 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.
Highlights
1.1 Basic setting and main resultIn this work, we study the higher Hölder regularity of solutions to nonlinear nonlocal equations of the formL A u = f in ⊂ Rn (1)driven by a kernel that potentially exhibits a very irregular behaviour
We study the higher Hölder regularity of solutions to nonlinear nonlocal equations of the form
By modifying an approach introduced in [2], we prove that so-called local weak solutions to such equations are locally Hölder continuous with some explicitly determined Hölder exponent
Summary
We study the higher Hölder regularity of solutions to nonlinear nonlocal equations of the form. Driven by a kernel that potentially exhibits a very irregular behaviour. By modifying an approach introduced in [2], we prove that so-called local weak solutions to such equations are locally Hölder continuous with some explicitly determined Hölder exponent. Supported by SFB 1283 of the German Research Foundation
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