Exponential Crystal Relaxation Model With P-Laplacian

TL;DR

This paper proves the global existence of weak solutions for a nonlinear exponential PDE with p-Laplacian, revealing a canceling effect between the exponential term and singular parts of the operator, extending previous linear case results.

Contribution

It establishes the existence of solutions for a time-dependent nonlinear PDE with exponential and p-Laplacian terms, including singular parts, for the first time.

Findings

Existence of weak solutions with singular parts in the p-Laplacian.

The canceling effect between exponential nonlinearity and singularities.

Precompactness achieved despite lack of time estimates.

Abstract

In this article we prove the global existence of weak solutions to an initial boundary value problem with an exponential and p-Laplacian nonlinearity. The equation is a continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation. In our investigation we find a weak solution where the exponent in the equation, $-\Delta_p u$, can have a singular part in accordance with the Lebesgue Decomposition Theorem. The singular portion of $-\Delta_p u$ corresponds to where $-\Delta_p u = -\infty$, which leads it to have a canceling effect with the exponential nonlinearity. This effect has already been demonstrated for the case of a linear exponent $p=2$, and for the time independent problem. Our investigation reveals that we can exploit this same effect in the time dependent case with nonlinear exponent. We obtain a solution by first forming a sequence of approximate solutions and then passing to the limit. The key to our existence result lies in the observation that one can still obtain the precompactness of the term $e^{-\Delta_p u}$ despite a complete lack of estimates in the time direction. However, we must assume that $1<p\leq 2$.