In this paper we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape. Using the shape sensitivities we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem. Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments.

@TechReport{BMW09,
  author       = {Burger, M. and Matevosyan, N. and Wolfram, M.-T.},
  title        = {A level set based shape optimization method for an elliptic obstacle problem},
  institution  = {WWU Muenster},
  year         = {2009},
  url          = \{/2009/BMW09},
}