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

Abstract

We prove local Hölder regularity for a nonlocal parabolic equations of the formfor p ∈ (1, ∞) and s ∈ (0, 1).

Hölder regularity for parabolic fractional $p$-Laplacian

“…1 After completion of the paper, we noticed a similar result in the recent preprint [1], via exponential change of variables.…”

“…Let u be a locally bounded, local weak sub(super)-solution to (1.1) -(1.3) in E T . Suppose that for some δ, σ and ξ in (0, 1 2 ), there holds…”

“…|K cR | for all t ∈ − aθR sp + θ(cR) sp , 0 , thanks to the arbitrariness of t. Given (5.16), we want to apply Lemma 3.4 with δ = 1, ξ = εξ 1 and ̺ = cR next. To this end, we first let ν be fixed in Lemma 3.1 (with δ = 1 ) and choose σ ∈ (0, 1 2 ) to satisfy…”

“…More generally, we find that the Hölder regularity is in fact encoded in a family of energy estimates in Corollary 2.1 and tools like logarithmic estimates or exponential change of variables play no role in our arguments. 1 As such, the arguments are new even for p = 2 and they hold the promise of a wider applicability, for instance, in Calculus of Variations.…”

Hölder regularity for mixed local and nonlocal $ p $-Laplace parabolic equations

“…Very recently, the Hölder regularity of weak solutions of (5) in the full range 1 < p < ∞ was investigated by Liao [27] without using any logarithmic estimate and any comparison principle. At the same time, Adimurthi-Prasad-Tewary [2] obtained the same result by employing the expansion of positivity and provided a new form of isoperimetric inequality. More results can be found in [7,25,20,24].…”