Abstract
We study a parabolic equation for the fractional p-Laplacian of order s, for pge 2 and 0<s<1. We provide space-time Hölder estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of Moser’s technique.
Highlights
We study the regularity of weak solutions to the nonlinear and nonlocal parabolic equation
The main result of our paper is the following Hölder regularity for local weak solutions of (1.1)
Definition of local weak solution, as well as of the spaces Cxδ,loc( × I ), we refer the reader to Sects. 3.1 and 2.3, respectively
Summary
We study the regularity of weak solutions to the nonlinear and nonlocal parabolic equation. |u(x) − u(x)|p u → RN ×RN |x − y|N +s p dx dy This operator can be seen as a nonlocal (or fractional) version of the p−Laplace operator,. (Homogeneity and scalings) It is important to notice that Eq (1.1) is not homogeneous, i.e. if u is a solution, λ u does not solve the same equation. We are concerned with the Hölder regularity for weak solutions of (1.1). To the best of our knowledge, our result is the first pointwise continuity estimate for solutions of this equation
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