Higher Hölder regularity for nonlocal equations with irregular kernel

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.

33 pages. Final version, to appear in Calc. Var. Partial Differential Equations

References in corpus (8)

• Local behavior of fractional $p$-minimizers
• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• The second eigenvalue of the fractional $p-$Laplacian
• Nonlocal equations with measure data
• On $L_p$-estimates for a class of non-local elliptic equations
• Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces
• Nonlinear commutators for the fractional p-Laplacian and applications
• Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel

Cited by in corpus (5)

• Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel
• Regularity estimates for fractional orthotropic $p$-Laplacians of mixed order
• On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations
• Convergence of level sets in fractional Laplacian regularization
• Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights