Failure of L^r-Calderón-Zygmund estimates for the p-Laplace equation for small r

2024/08/07 by Schikorra, Armin · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.03546

Abstract

Let p ≠ 2. For any small enough r> max \p-1,1\ and for any Λ> 1 there exists a Lipschitz function u and a bounded vectorfield f such that \begincases \rm div(|∇ u|^p-2 ∇ u) = \rm div (f) \quadamp; \textin \mathbbB²
u=0 amp;\texton ∂ \mathbbB² \endcases but ∫_\mathbbB² |∇ u|^r \not ≤ Λ∫_\mathbbB² |f|^(r)/(p-1). This disproves a conjecture by Iwaniec from 1983. The proof adapts recent convex-integration ideas by Colombo-Tione.

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• Non-uniqueness and failure of Calderón-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth
• Sacks-Uhlenbeck type regularity for subcritical generalized p-harmonic maps into Homogeneous targets

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