C1, regularity for the normalized p-Poisson problem

Abstract

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Gradient continuity estimates for the normalized p-Poisson equation

“…This improves one of the regularity results in [3], where a C 1,α estimate was established depending on the L m norm of f under the additional restriction that p > 2 and m > max(2, n, p 2 ) (see Theorem 1.2 in [3]). We also mention that differently from the approach in [3], which uses methods from divergence form theory and nonlinear potential theory in the proof of Theorem 1.2, our method is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [36]. Moreover, for f continuous, our approach also gives a somewhat different proof of the C 1,α regularity result, Theorem 1.1, in [3].…”

“…Therefore, in order to obtain improvement of flatness at each scale after a rescaling, it is imperative to get uniform C 1 −type estimates independent of |p 0 | for the limiting equations corresponding to the case f ≡ 0. This is precisely done in [3] by an adaptation of the Ishii-Lions approach as in [20], where the authors obtained uniform Lipschitz estimates for solutions to (1.8) for large |p 0 | ′ s when f = 0. In this paper, we follow an approach which is different from that in [3].…”

“…This is precisely done in [3] by an adaptation of the Ishii-Lions approach as in [20], where the authors obtained uniform Lipschitz estimates for solutions to (1.8) for large |p 0 | ′ s when f = 0. In this paper, we follow an approach which is different from that in [3]. Our proofs of Theorem 2.2 and Theorem 2.3 are based instead on separation of the degenerate and the non-degenerate phase, and do not rely on the uniform Lipschitz estimates for equations of the type (1.8) for large |p 0 | ′ s. This is inspired by ideas in [36], where an alternate proof of C 1,α −regularity for the p−Laplacian was given.…”

Equivalence of viscosity and weak solutions for the normalized p(x)-Laplacian

“…Recently the equivalence of solutions has been studied for various equations. These include the normalized p-Poisson equation [APR17], a non-homogeneous p-Laplace equation [MO] and the normalized p(x)-Laplace equation [Sil18]. Moreover, in [PV] the equivalence is shown for the radial solutions of a parabolic equation.…”

Sharp regularity estimates for quasilinear evolution equations

“…We then iterate this estimate in a systematic manner, properly adjusted to the intrinsic scaling of equation. Inspired by the recent results from [5,6,8] and [39], we obtain an estimate (Theorem 3.2), which provides a precise control of the oscillation of weak solution of (1.1) in terms of the magnitude of its gradient.…”