Shape optimization for non-linear parabolic problems on surfaces

Abstract

Shape optimization is concerned with finding optimal shapes in the sense that they minimize a cost functional, most often while satisfying one or more constraints. This work considers a problem in the following context: The set of admissible shapes is the set of all closed, compact and smooth surfaces embedded in R3. The cost functional is the squared L2 norm of u-ud over the surface for a target function ud. The function u is given by the constraint and is the solutionof a non-linear parabolic partial dierential equation on the surface. While elliptic problems as well as problems on open domains are widely discussed, the studying of parabolic problems onsurfaces is a rather new field of research. Existence and uniqueness of a solution of both the optimization problem and the constraintare discussed. The shape derivative is calculated both in a general context as well as for theconcrete example of a type of reaction-diusion equation with applications in cell biology. A gradient descent algorithm via the shape derivative is deducted, an implementation is givenand its results are reviewed. Moreover, a technique to guarantee surface area preservation isintroduced. The necessary theoretical background, both from dierential geometry as well asfrom the theory of partial dierential equations, is presented and applied to the problem athand.