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arXiv · 2609.12180

Optimal regularity for vectorial, higher order and non-minimizing Bernoulli problems

Guido De Philippis·Jonas Hirsch·Mickaël Nahon
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Abstract

We prove the Lipschitz regularity of solutions for a wide class of generalized Bernoulli free boundary problems, in vectorial, higher order, and non-minimizing settings. Our method does not rely on notions of viscosity solutions or comparison methods, which allows us to reach the optimal regularity for Bernoulli-type problems associated to elliptic operators which do not satisfy any maximum principle, namely the elasticity, biharmonic and Stokes equation. With a similar method we obtain the optimal Lipschitz regularity for stationary, non-minimizing solutions of the standard Bernoulli problem in two dimensions.

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Guido De Philippis, Jonas Hirsch, Mickaël Nahon. 2026-09-10. Optimal regularity for vectorial, higher order and non-minimizing Bernoulli problems. https://arxiv.org/abs/2609.12180

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