Lipschitz Regularity of Fractional p-Laplacian

Abstract

In this article, we investigate the Holder regularity of the fractional p-Laplace equation of the form (-triangle p)(s) u = f where p > 1, s is an element of (0, 1) and f is an element of L-loc(infinity) (ohm). Specifically, we prove that u is an element of C-degrees loc(0, gamma) (ohm) for gamma(degrees) = min{1, sp/p- 1}, provided that sp/p- 1 not equal 1. In particular, it shows that u is locally Lipschitz for sp/p- 1 > 1. Moreover, we show that for sp/p-1 = 1, the solution is locally Lipschitz, provided that f is locally Holder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems.