Height estimates for exterior problems of capillarity type

Abstract

This work concerns boundary value problems for a class of nonlinear equations modeled on the physical equations for a capillary free surface in a gravitational field. The results consist principally of estimates for the height of a solution in an exterior domain. Structure conditions reflecting the nonlinearity of the mean curvature operator are imposed on a class of symmetric variational operators in terms of the Legendre transform of the variational integrand. Estimates are found for the boundary height of a rotationally symmetric solution in the exterior of a ball of radius R. These estimates, which are valid for any R9 are shown to be asymptotically exact as R tends to zero or infinity. This provides a proof of the asymptotic behavior of the boundary height which previously has been derived by a formal perturbation method. An asymptotic characterization of the solution in a neighborhood of the boundary is also given. For a general domain estimates are obtained from a maximum principle due to Finn in which the symmetric solutions serve as comparison functions. l Preliminaries* We begin by formulating the standard capillarity problem on an exterior domain n dimensional space; the physical case is n = 2. Let Ω c Rn be an exterior domain whose boundary, Σ, is a compact C1 hypersurface. When the generalized (vertical) cylinder Σ x R is immersed in an infinite reservoir of fluid, the action of capillarity gives rise to a free surface of static equilibrium in the outside of the cylinder. Let the height of this capillary free surface, assumed nonparametric over Ω, be given by the scalar function u(x)f x — (xlf - , xn) eΩ. Physical principles assert that the equilibrium free surface minimizes the functional