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We consider a shape optimization problem for an elliptic partial differential equation with uncertainty in its input data. The design variable enters the lower-order term of the state equation and is modeled through the characteristic function of a measurable subset of the spatial domain. As usual, a measure constraint is imposed on the design variable. In order to compute a robust optimal shape, the objective function involves a weighted sum of both the mean and the variance of the compliance. Since the optimization problem is not convex, a full relaxation of it is first obtained. The relaxed problem is then solved numerically by using a gradient-based optimization algorithm. To this end, the adjoint method is used to compute the continuous gradient of the cost function. Since the variance enters the cost function, the underlying adjoint equation is non-local in the probabilistic space. Both the direct and adjoint equations are solved numerically by using a sparse grid stochastic collocation method. Three numerical experiments in 2D illustrate the theoretical results and show the computational issues which arise when uncertainty is quantified through random fields.

Robust optimal shape design for an elliptic PDE with uncertainty in its input data