Partial regularity for an exponential PDE in crystal surface models

TL;DR

This paper establishes partial regularity results for solutions to a PDE modeling crystal surfaces, showing that the exponential term remains bounded on a large subset of the domain and describing the behavior near singular points.

Contribution

It provides the first partial regularity result for a PDE with an exponential term derived from crystal surface models, addressing singularity issues.

Findings

Existence of an open subset where the exponential term is locally bounded.

At points outside this subset, the solution vanishes at a specific polynomial rate.

The exponential term behaves well away from negative infinity.

Abstract

We study the regularity properties of a weak solution to the boundary value problem for the equation $-\Delta \rho +a u=f$ in a bounded domain $\Omega\subset \mathbb{R}^N$, where $\rho=e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u+\beta_0|\nabla u|^{-1}\nabla u\right)}$. This problem is derived from the mathematical modeling of crystal surfaces. It is known that the exponent term can exhibit singularity. In this paper we obtain a partial regularity result for the weak solution. It asserts that there exists an open subset $\Omega_0\subset \Omega$ such that $|\Omega\setminus\Omega_0|=0$ and the exponent term is locally bounded in $\Omega_0$. Furthermore, if $x_0\in \Omega\setminus\Omega_0$, then $\rho$ vanishes of $N+2-\varepsilon$ order at $x_0$ for each $\varepsilon\in(0,2)$. Our results reveal that the exponent term behaves well if it stays away from negative infinity.