A p-Laplacian heat equation in a non-cylindrical domain with an oscillating boundary: A homogenization process

Abstract

This article addresses the homogenization of the heat equation involving the $p$-Laplacian in non-cylindrical domains with an evolving oscillating boundary. A change of coordinates is employed to transform the heat equations with $p$-Laplacian into parabolic $p$-Laplacian equations featuring oscillating coefficients in a reference domain. One novelty of this article is that the equation in the reference domain consists of an oscillating coefficient matrix in the nonlinear component, specifically ${\left|M_{\epsilon }^{tr}\nabla U_{\epsilon }\right|}^{p-2}$. The existence and uniqueness of solutions are demonstrated in the reference domain through a non-trivial Galerkin approximation, accompanied by a significant $\epsilon$-uniform estimate. On the other hand, a modified two-scale convergence method is employed to derive the two-scale homogenized problem. Furthermore, an explicit solution to the nonlinear cell problem is constructed. This solution is employed to drive the effective equation within the reference domain and corrector result, identified as a transformed effective problem of the heat equation with $p$-Laplacian in a non-cylindrical domain featuring an effective evolving boundary.