Optimal Control of Free Boundary Problems with Surface Tension Effects
We consider a PDE constrained optimization problem governed by a free boundary problem. The state system consists of 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. This amounts to solving a second order system both in the bulk and on the interface. Our analysis hinges on a convex constraint on the control which always enforce the state constraints. Using only first order regularity we show that the control-to-state operator is twice Frechet differentiable. We improve slightly the regularity of the state variables and exploit this to show existence of a control together with second order sufficient optimality conditions. We prove that the state and adjoint systems have the requisite regularity for the error analysis (strong solutions). We discretize the state, adjoint and control variables via piecewise linear finite elements and show optimal first order error estimates for all variables, including the control. We conclude with a more realistic model governed by the Stokes equations in the bulk and slip boundary conditions on the free boundary: we deal with minimal Sobolev regularity of the domain boundary.
This work is joint with H. Antil and P. Sodre.