We prove some Bernstein-type theorems for a class of stationary points of the Alt–Caffarelli functional in and arising as limits of the singular perturbation problem
in the unit ball as . Here , , , is an approximation of the Dirac measure and . The limit functions of uniformly converging sequences solve a Bernoulli-type free boundary problem in some weak sense. Our approach has two novelties: First we develop a hybrid method for stratification of the free boundary of blow-up solutions which combines some ideas and techniques of viscosity and variational theory. An important tool we use is a new monotonicity formula for the solutions based on a computation of J. Spruck. It implies that any blow-up of either vanishes identically or is a homogeneous function of degree 1, that is, , , in spherical coordinates . In particular, this implies that in two dimensions the singular set is empty at the nondegenerate points, and in three dimensions the singular set of is at most a singleton. Second, we show that the spherical part is the support function (in Minkowski’s sense) of some capillary surface contained in the sphere of radius . In particular, we show that is an almost conformal and minimal immersion and the singular Alt–Caffarelli example corresponds to a piece of catenoid which is a unique ring-type stationary minimal surface determined by the support function .
"Capillary surfaces arising in singular perturbation problems." Anal. PDE 13 (1) 171 - 200, 2020. https://doi.org/10.2140/apde.2020.13.171