Regularity of the Free Boundary in Bernoulli-Type Problems
Time: 2026-04-27
Published By: Qiuye Huang
Speaker(s): Morteza Fotouhi Firoozabad (University of Nottingham Ningbo China)
Time: 15:00-16:00 April 27, 2026
Venue: Room78201, Jingchunyuan 78, BICMR
Abstract: Minimizers of the functional
$$\mathcal{J}(u) = \int_\Omega |\nabla u|^p + (p-1)\Lambda^p \mathbf{1}_{\{u>0\}} \, dx
$$
give rise to the Bernoulli free boundary problem for the $p$-Laplacian. Such minimizers solve the overdetermined system
$$
\Delta_p u = \operatorname{div}\big(|\nabla u|^{p-2}\nabla u\big) = 0 \ \text{ in } \{u>0\}, \qquad u = g \ \text{ on } \partial\Omega, \qquad |\nabla u| = \Lambda \ \text{ on } \Gamma:=\partial\{u>0\},
$$
where the interface $\Gamma$ is not prescribed a priori and is therefore called the free boundary. Understanding the regularity of $\Gamma$ is a central challenge in the theory of free boundary problems.
In this talk I will first review classical and recent regularity results for the one-phase Bernoulli problem, including local Lipschitz regularity and $C^{1,\alpha}$ regularity of the free boundary away from a small singular set. I will then turn to the two-phase setting, where the solution is allowed to change sign, and present results showing that minimizers are locally Lipschitz continuous and that the free boundary at two-phase points is $C^{1,\alpha}$ regular.
