Borderline gradient continuity for the normalized $p$-parabolic operator

TL;DR

This paper establishes borderline gradient continuity estimates for solutions to the normalized p-parabolic operator, extending previous results and providing new regularity results based on the integrability of the source term.

Contribution

It introduces novel gradient continuity estimates for viscosity solutions of the normalized p-parabolic equation using a parabolic Riesz potential, independent of the Ishii-Lions method.

Findings

Proves gradient continuity estimates in terms of the critical $L(n+2,1)$ norm of $f$.

Establishes H"older continuity of the spatial gradient for $f otin L^{n+2}$.

Improves upon previous gradient regularity results for bounded source terms.

Abstract

In this paper, we prove gradient continuity estimates for viscosity solutions to $\Delta_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm of $f$, where $\Delta_{p}^N$ is the game theoretic normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for $u$ in terms of the modified parabolic Riesz potential $\mathbf{P}^{f}_{n+1}$ as defined in (2.8) below. Moreover, for $f \in L^{m}$ with $m>n+2$, we also obtain H\"older continuity of the spatial gradient of the solution $u$, see Theorem 2.6 below. This improves the gradient H\"older continuity result in [3] which considers bounded $f$. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in [9]. Moreover differently from that in [3], our approach is independent of the Ishii-Lions method which is crucially used in [3] to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.