Optimal Control of a Free Boundary Problem: Analysis with Second-Order Sufficient Conditions

Abstract

We consider a PDE-constrained optimization problem governed by a free boundary problem. The state system is based on coupling the Laplace equation in the bulk with a Young--Laplace equation on the free boundary to account for surface tension, as proposed by P. Saavedra and L. R. Scott [Math. Comp., 57 (1991), pp. 451--475]. This amounts to solving a second order system both in the bulk and on the interface. Our analysis hinges on a convex control constraint such that the state constraints are always satisfied. Using only first-order regularity we show that the control-to-state operator is twice continuously Fréchet differentiable. We improve slightly the regularity of the state variables and exploit it to show existence of a control together with second- order sufficient optimality conditions.