On the supremal version of the Alt-Caffarelli minimization problem (Q2039524)

The authors consider the free boundary problem (P)\(_\Lambda\): \[m_\Lambda:=\min\{J_\Lambda(u):=\|\nabla u\|_\infty+\Lambda|\{u>0\}|:u\in \text{Lip}_1(\Omega)\},\] where \(\Lambda>0,\ |\{u>0\}|\) is the Lebesgue measure of the set \(\{x\in \Omega:u(x)>0\}\) and \[\text{Lip}_1(\Omega):= \{u\in W^{1,\infty}(\Omega)\,:u\ge 0\ \text{in}\ \Omega, \ u=1\ \text{on}\ \partial\Omega\}\] The main results include the existence and uniqueness of the non-constant solution on convex domains, the identification and the geometrical characterization of the variational infinity Bernoulli constant \[\Lambda_{\Omega,\infty}:=\inf\{\Lambda>0:\,(P)_\Lambda\ \text{admits a non-constant solution}\}.\]