Surface Tension Free Boundary Problems: Formulation, Optimal Control and Numerics

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Carlos, Patrick Sodré
Nochetto, Ricardo H
The goal of this work is to treat the formulation, optimal control and numerical analysis of free boundary problems with surface tension effects. From a formulation point of view, we introduce a (dimension independent) abstract framework which captures the essential behavior of free boundary problems with surface tension effects. We then apply this framework to two scenarios. The first is where the underlying bulk system is governed by the Laplacian with non-homogeneous essential boundary condition, and the second is modeled by the Stokes equations with slip and no-slip boundary conditions. We do not impose a fixed contact angle between the free surface and any fixed part of the boundary. Although the formulation and numerics involving the Laplacian was available in the literature, the Stokes free boundary problem in R<super>n</super> is novel. To obtain this last result we also had to prove the existence and uniqueness in Sobolev spaces for the pure slip problem for domains of type C<super>1,\epsilon</super>. This is a significant improvement over the current best result involving C<super>1,1</super> domains. The results from the abstract formulation also carry over to the optimal control aspect. We obtain differentiability conditions which guarantee existence and (local) uniqueness of a minimizer to well-behaved cost functions. In the Laplacian case we go beyond the theoretical results and give precise second-order sufficient conditions for the (local) uniqueness of a minimizer for cost functions of the tracking type. The contribution in this area is significant in the sense that sufficient conditions are usually only assumed to be true, while we actually show that it indeed holds for our specific problem. The last piece of this work is the numerical treatment of the free boundary optimal control problem based on the Laplace equation. We are able to prove optimal convergence results using the finite element method. Moreover, we construct experiments to study the behavior of various metrics associated with the optimization problem.