William M. Feldman

Dynamic Stability of Equilibrium Capillary Drops

We investigate a model for contact angle motion of quasi-static capillary drops resting on a horizontal plane. We prove global in time existence and long time behavior (convergence to equilibrium) in a class of star-shaped initial data for which we show that topological changes of drops can be ruled out for all times. Our result applies to any drop which is initially star-shaped with respect to a small ball inside the drop, given that the volume of the drop is sufficiently large. For the analysis, we combine geometric arguments based on the moving-plane type method with energy dissipation methods based on the formal gradient flow structure of the problem.

Stability of Serrin's Problem and Dynamic Stability of a Model for Contact Angle Motion

We study the quantitative stability of Serrins symmetry problem and its connection with a dynamic model for contact angle motion of quasi-static capillary drops. We prove a new stability result which is both linear and depends only on a weak norm \[ \big\||Du|^2- 1\big\|_{L^2(\partial \Omega)}. \] This improvement is particularly important to us since the

A Free Boundary Problem with Facets

L^2(\partial \Omega)

Homogenization of the oscillating Dirichlet boundary condition in general domains

norm squared of

Homogenization and non-homogenization of certain non-convex Hamilton–Jacobi equations

|Du|^2-1

Continuity and discontinuity of the boundary layer tail

is exactly the energy dissipation rate of the associated dynamic model. Combining the energy estimate for the dynamic model with the new stability result for the equilibrium problem yields an exponential rate of convergence to the steady state for regular solutions of the contact angle motion problem. As far as we are aware this is one of the first applications of a stability estimate for a geometric minimization problem to show dynamic stability of an associated gradient flow.

Quantitative homogenization of elliptic partial differential equations with random oscillatory boundary data

We study a free boundary problem on the lattice whose scaling limit is a harmonic free boundary problem with a discontinuous Hamiltonian. We find an explicit formula for the Hamiltonian, prove that the solutions are unique, and prove that the limiting free boundary has a facets in every rational direction. Our choice of problem presents difficulties that require the development of a new uniqueness proof for certain free boundary problems. The problem is motivated by physical experiments involving liquid drops on patterned solid surfaces.