On the Viscosity Solutions of Parabolic p-Laplacian Equations with Capillary-Type Boundary Conditions

TL;DR

This paper proves well-posedness and analyzes the long-term behavior of viscosity solutions to parabolic p-Laplacian equations with capillary boundary conditions on convex domains, using gradient estimates and approximation techniques.

Contribution

It establishes existence, uniqueness, and regularity results for solutions to singular/degenerate parabolic p-Laplacian equations with capillary boundary conditions, including new regularity and asymptotic analysis.

Findings

Established gradient estimates independent of the solution's C^0 norm.

Proved existence and uniqueness of solutions.

Derived sharp regularity results and asymptotic behavior.

Abstract

In this paper, we establish the well-posedness and large-time asymptotic behavior of viscosity solutions to singular/degenerate parabolic $p$-Laplacian equations with general capillary-type boundary conditions, including Neumann and prescribed contact angle cases, on strictly convex domains. By establishing a gradient estimate independent of the $C^0$ norm of the solution via the maximum principle, and by analyzing the problem through an approximation procedure together with associated elliptic eigenvalue problems, we prove the existence, uniqueness, and asymptotic behavior of solutions. For the elliptic problem with Neumann boundary conditions, we first focus on flat domains with the zero Neumann condition. By reflecting $u$ across the flat boundary $T_1$ and then using inf- and sup-convolution arguments in the reflected domain, we obtain the $C^{1,\alpha}$ result. For the general elliptic case, we obtain sharp global $C^{1,\alpha}$ regularity by flattening the boundary and employing compactness arguments together with an ``improvement of flatness'' iteration. With an extra condition in the iteration, we can also deal with the singular case $1<p<2$. In the parabolic setting, the spatial H\"older regularity of $Du$ follows from elliptic estimates combined with the Lipschitz continuity of $u$ in time, which in turn yields joint H\"older continuity in $(x,t)$. Extensions to non-convex domains are also discussed by incorporating a suitable forcing term.