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arXiv · 2402.04951

Gradient continuity for the parabolic $(1,\,p)$-Laplace equation under the subcritical case

Shuntaro Tsubouchi
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Abstract

This paper is concerned with the gradient continuity for the parabolic $(1,\,p)$-Laplace equation. In the supercritical case $\frac{2n}{n+2} n(2-p)/p$. The gradient continuity is proved, similarly to the supercritical case, once the local gradient bounds of solutions are verified. Hence, this paper mainly aims to show the local boundedness of a solution and its gradient by Moser's iteration. The proof is completed by considering a parabolic approximate problem, and showing a priori gradient estimates of a bounded weak solution to the relaxed equation.

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Shuntaro Tsubouchi. 2024-02-07. Gradient continuity for the parabolic $(1,\,p)$-Laplace equation under the subcritical case. https://arxiv.org/abs/2402.04951

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