Calculus of Variations and Geometric Measure Theory

M. Colombo - R. Tione

Non-classical solutions of the $p$-Laplace equation

created by tione on 20 Jan 2022
modified on 04 Nov 2024

[BibTeX]

Published Paper

Inserted: 20 jan 2022
Last Updated: 4 nov 2024

Journal: JEMS
Year: 2022

ArXiv: 2201.07484 PDF

Abstract:

In this paper we answer Iwaniec and Sbordone's conjecture \cite{IB94} concerning very weak solutions to the $p$-Laplace equation. Namely, on one hand we show that distributional solutions of the $p$-Laplace equation in $W^{1,r}$ for $p \neq 2$ and $r>\max\{ 1,p-1\}$ are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration non-energetic distributional solutions if this cone condition is not met, thus answering negatively Iwaniec and Sbordone's conjecture in general.