Non-uniqueness and failure of Calder\'on-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth

Akshara Vincent

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arXiv · 2510.20024

Non-uniqueness and failure of Calder\'on-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth

Akshara Vincent

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Abstract

We provide counterexamples to uniqueness of solutions as well as a priori Calder\'on-Zygmund estimates for solutions below $L^2$ using convex integration argument for equations of the type $$ \text{div} (A (\nabla u)) = 0 \quad \text{in } \mathbb{B}^2, $$ where $A: \mathbb{R}^{2} \to \mathbb{R}^2$ is smooth, uniformly elliptic and has essentially linear growth, but fails to be monotone and asymptotically Uhlenbeck.

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Akshara Vincent. 2025-10-22. Non-uniqueness and failure of Calder\'on-Zygmund estimates below the critical exponent for non-monotone PDE with linear growth. https://arxiv.org/abs/2510.20024

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