Continuity of the trace operator with respect to the domain and application to shape optimization

Shape optimization amounts to find the optimal shape of a domain which minimizes a given criterion, often called a cost functional. Here, we are interested in the case where the criterion is computed through the solution of a partial differential equation, the so-called state equation, which makes the optimization problem non-trivial. We use a general parameterization of the unknown boundary in order to preserve the physical general information and we prove the continuity of the cost functional