Lipschitz regularity for parabolic double phase equations with gradient nonlinearity

2025/08/22 by Sen, Abhrojyoti, Siltakoski, Jarkko · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.16391

Abstract

We establish the local Lipschitz regularity in space for the viscosity solutions to the parabolic double phase equation of the form \smash∂_tu-div (|Du|^p-2D u+a(z)|D u|^q-2D u)=f(z, Du) by employing the Ishii-Lions method. In addition, we obtain Hölder estimate in time which turns out to be sharp in the degenerate regime. Here, 1< p≤ q<∞, and the coefficient a≥ 0 is assumed to be bounded, locally Lipschitz continuous in space, and continuous in time. Furthermore, the non-homogeneity f is assumed to be continuous on Ω× ℝ× ℝ^N, and to satisfy a suitable gradient growth condition. We also establish the equivalence between bounded viscosity solutions and weak solutions, under appropriate additional regularity assumption on the coefficient a.

Cited by

• Gradient higher integrability of bounded solutions to parabolic double-phase systems

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