CAPILLARY SURFACES ARISING IN SINGULAR PERTURBATION PROBLEMS

We prove some Bernstein-type theorems for a class of stationary points of the Alt-Caffarelli functional in R-2 and R-3 arising as limits of the singular perturbation problem {Delta u(epsilon)(x) = beta(epsilon)(u(epsilon)) in B-1, vertical bar u(epsilon)vertical bar <= 1 in B-1, in the unit ball B-1 as epsilon -> 0. Here beta(epsilon)(t) = (1/epsilon)beta(t/epsilon) >= 0, beta is an element of C-0(infinity)[0, 1], integral(1)(0) beta(t) dt = M > 0, is an approximation of the Dirac measure and epsilon > 0. The limit functions u = lim(epsilon j -> 0) u(epsilon j) of uniformly converging sequences {u(epsilon j)} 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 partial derivative{u(0) > 0} 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 u(epsilon) based on a computation of J. Spruck. It implies that any blow-up u(0) of u either vanishes identically or is a homogeneous function of degree 1, that is, u(0) = rg(sigma), sigma is an element of SN-1, in spherical coordinates (r, theta). 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 u(0) is at most a singleton. Second, we show that the spherical part g is the support function (in Minkowski's sense) of some capillary surface contained in the sphere of radius root 2M. In particular, we show that del u(0) : S-2 -> R-3 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 g.