Non-classical solutions of the p-Laplace equation

2022/01/19 by Maria Colombo, Colombo, Maria, Riccardo Tione +1 · 4 citations
Mathematics · #Mathematical and Theoretical Analysis #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2201.07484

Abstract

In this paper we answer Iwaniec and Sbordone's conjecture \citeIB94 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 ≠ 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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