Local regularity for nonlocal equations with variable exponents

Abstract

In this paper, we study local regularity properties of minimizers of nonlocal variational functionals with variable exponents and weak solutions to the corresponding Euler--Lagrange equations. We show that weak solutions are locally bounded when the variable exponent $p$ is only assumed to be continuous and bounded. Furthermore, we prove that bounded weak solutions are locally Hölder continuous under some additional assumptions on $p$. On the one hand, the class of admissible exponents is assumed to satisfy a log-Hölder-type condition inside the domain, which is essential even in the case of local equations. On the other hand, since we are concerned with nonlocal problems, we need an additional assumption on $p$ outside the domain.

28 pages, 1 figure. corrected typos, fixed the proof of Theorem 1.5, added two references

References in corpus (1)

• Robin fractional problems with symmetric variable growth

Cited by in corpus (1)

• Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights