Gradient continuity estimates for the normalized p-Poisson equation

2019/04/30 by Banerjee, Agnid, Munive, Isidro H. · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.13076

Abstract

In this paper, we obtain gradient continuity estimates for viscosity solutions of Δ_p^N u= f in terms of the scaling critical L(n,1 ) norm of f, where Δ_p^N is the normalized p-Laplacian operator defined in (1.2) below. Our main result, Theorem 2.2, corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential I^f_q. Moreover, for f ∈ L^m with m>n, we also obtain C^1,α estimates, see Theorem 2.3 below. 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].

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