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

This paper is concerned with the gradient continuity for the parabolic \((1,\,p)\)-Laplace equation. In the supercritical case \(\frac{2n}{n+2}<p<\infty \), where \(n\ge 2\) denotes the space dimension, this gradient regularity result has been proved recently by the author. In this paper, we would like to prove that the same regularity holds even for the subcritical case \(1<p\le \frac{2n}{n+2}\) with \(n\ge 3\), on the condition that a weak solution admits the \(L^{s}\)-integrability with \(s>\frac{n(2-p)}{p}\). 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, verifying a comparison principle, and showing a priori gradient estimates of a bounded weak solution to the relaxed equation.