PreprintCC BY 4.0HAL
Optimal regularity for vectorial, higher order and non-minimizing Bernoulli problems
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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BibTeX
@misc{philippis2026optimal,
title = {{Optimal regularity for vectorial, higher order and non-minimizing Bernoulli problems}},
author = {Guido De Philippis and Jonas Hirsch and Mickaël Nahon},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12180},
eprint = {2609.12180},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05746140v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal regularity for vectorial, higher order and non-minimizing Bernoulli problems
AU - Guido De Philippis
AU - Jonas Hirsch
AU - Mickaël Nahon
PY - 2026
DA - 2026/09/29
PB - HAL
DO - 10.48550/arxiv.2609.12180
UR - https://hal.science/hal-05746140v2
AB - 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.
ER -