Borderline gradient continuity for the normalized p-parabolic operator

2022/11/28 by Akman, Murat, Banerjee, Agnid, Munive, Isidro H.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2211.15246

Abstract

In this paper, we prove gradient continuity estimates for viscosity solutions to Δ_p^N u- u_t= f in terms of the scaling critical L(n+2,1 ) norm of f, where Δ_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 P^f_n+1 as defined in (2.8) below. Moreover, for f ∈ L^m with m>n+2, we also obtain Hölder continuity of the spatial gradient of the solution u, see Theorem 2.6 below. This improves the gradient Hölder 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.

Related

• Gradient continuity estimates for the normalized p-Poisson equation
• Improved regularity for the parabolic normalized p-Laplace equation
• Gradient Holder regularity for parabolic normalized p(x,t)-Laplace equation
• Hölder regularity for the gradient of the inhomogeneous parabolic normalized p-Laplacian
• Boundary Hölder gradient estimates for parabolic p-Laplace type equations