Remarks on regularity for p-Laplacian type equations in non-divergence form

Abstract

We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the p-Laplacian type and in non-divergence form. We provide local Hölder and Lipschitz estimates for the solutions. In the degenerate case, we prove the Hölder regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.

Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations

“…We refer to [13] and [18] for existence and uniqueness of viscosity solutions of (1.2)-(1.3). For the reader's convenience, in Appendix (Section A.1), we list more precise structure assumptions on the F λ besides (1.9), which guarantee the comparison principle; see more details also in [ [3,45,46,36,5,4,2] and references therein.…”

A mean value formula for the variational p-Laplacian

“…Case 3: Let ∇ϕ(x 0 ) = 0 and assume that u is not constant in any ball B ρ (x 0 ). Then we may argue as in the proof of Proposition 2.4 in [4] to prove that there is a sequence y k → 0 such that the functions ϕ k (x) = ϕ(x + y k ) touches u from above at points x k and ∇ϕ k (x k ) = 0 for all k. As in Case 1, we get…”

“…As expected, when p ≥ 2 or ∇φ(x) = 0, this definition is equivalent to the usual one (cf. [4]). This definition of viscosity solution adds some extra technicalities in the proofs of this manuscript.…”

Gradient continuity estimates for the normalized p-Poisson equation

“…This aspect was first studied in [33] . In recent times, the parabolic normalized p−Laplacian, as well as its degenerate and singular variants, have been studied in various contexts in a number of papers, see [1,22,13,5,6,7,18,32,19,23,21,31]. Such equations have also found applications in image processing (see for instance [13]).…”