Non-existence of cusps for degenerate Alt–Caffarelli functionals

Source 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)= \mathop\mathrm{dist}\nolimits(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.