Hölder regularity for fractional $p$-Laplace equations

Abstract

We give an alternative proof for Hölder regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by a continuous iteration. Moreover, we also obtain a De Giorgi type isoperimetric inequality for all s ∈ (0, 1), which partially answers a question of M.Cozzi.

Local Hölder regularity for nonlocal parabolic $p$-Laplace equations

“…through a logarithmic estimate [18]. See [14,1] for discussions on fractional versions of De Giorgi isoperimetric inequalities.…”

“…The subsequent work of Cozzi [13] covered a stable (in the limit s → 1) proof of Hölder regularity by defining a novel fractional DeGiorgi class. An alternate proof of Hölder regularity based on a differential inequality was given in [1]. Explicit exponents for Hölder regularity were found in [6,7].…”

“…One of the main difficulties we face when dealing with regularity issues for nonlocal equations is the lack of a corresponding isoperimetric inequality for W s,p functions. Indeed, since such functions can have jumps, a generic isoperimetric inequality seems out of reach at this time (see [14,1] ) One way around this issue is to note that since we are working with solutions of an equation, which we expect to be continuous and hence have no jumps, we could try and cook up an isoperimetric inequality for solutions. Such a strategy turns out to be feasible due to the presence of the "good term" or the isoperimetric term in the Caccioppoli inequality.…”