We study a class of shape optimization problems for semi-linear elliptic equations with Dirichlet boundary conditions in smooth domains in ℝ 2 . A part of the boundary of the domain is variable as the graph of a smooth function. The problem is equivalently reformulated on a fixed domain. Continuity of the solution to the state equation with respect to domain variations is shown. This is used to obtain differentiability in the general case, and moreover a useful formula for the gradient of the cost functional in the case where the principal part of the differential operator is the Laplacian.
Applied Mathematics and Optimization – Springer Journals
Published: Apr 1, 2004
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