$q$-Moment Estimates for the Singular $p$-Laplace Equation and Applications

TL;DR

This paper establishes $q$-moment estimates and uniform integrability for solutions of the singular $p$-Laplace equation, leading to mass conservation, weak convergence, and convergence rates in Wasserstein distance.

Contribution

It introduces new $q$-moment estimates and convergence results for the singular $p$-Laplace equation, expanding understanding of solution behavior in critical parameter ranges.

Findings

Derived $q$-uniform integrability for certain parameters

Established mass conservation and weak convergence results

Provided a convergence rate of order $t^{q-1}$ in Wasserstein distance

Abstract

We provide $q$-moment estimates on annuli for weak solutions of the singular $p$-Laplace equation where $p$ and $q$ are conjugates. We derive $q$-uniform integrability for some critical parameter range. As a application, we derive a mass conservation as well as a weak convergence result for a larger critical parameter range. Concerning the latter point, we further provide a rate of convergence of order $t^{q-1}$ of the solution in the $q$-Wasserstein distance.