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

Maria Colombo; Riccardo Tione

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arXiv · 2201.07484

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

Maria Colombo·Riccardo Tione

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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.

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Maria Colombo, Riccardo Tione. 2022-01-19. Non-classical solutions of the $p$-Laplace equation. https://arxiv.org/abs/2201.07484

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