Non-existence of cusps for degenerate Alt–Caffarelli functionals

Abstract

We study a class of free-boundary problems for degenerate one-phase Alt–Caffarelli functionals $J_Q(v,\Omega ):={\int }_{\Omega }|\nabla v{|}^2+Q^2(x){\chi }_{\{v>0\}}dx$. More specifically, we consider $Q(x)=\mathrm{dist}(x,\Gamma {)}^{\gamma }$ for an affine $k$-plane $\Gamma$ and $\gamma >0$. Because $Q$ vanishes on $\Gamma$, the techniques of Alt and Caffarelli (1981) for proving non-degeneracy of local minimizers $u$ and weak geometric regularity of their positivity sets $\{u>0\}$ fail near $\Gamma$. In this note we prove that despite the degeneration of $Q$ local minimizers, $u$ and $\{u>0\}$ still satisfy an analogous non-degeneracy condition near $\Gamma$. This non-degeneracy is sufficient to eliminate the existence of cusps in a wider class of cases than previously known.