Viscosity solutions for a model of contact line motion

Abstract

This paper considers a free boundary problem that describes the motion of contact lines of a liquid droplet on a flat surface. The elliptic nature of the equation for droplet shape and the monotonic dependence of contact line velocity on contact angle allows us to introduce a notion of “viscosity” solutions for this problem. Unlike similar free boundary problems, a comparison principle is only available for a modified short-time approximation because of the constraint that conserves volume. We use this modified problem to construct viscosity solutions to the original problem under a weak geometric restriction on the free boundary shape. We also prove uniqueness provided there is an upper bound on front velocity.