Shape sensitivity for the Laplace–Beltrami operator with singularities

This paper considers shape sensitivity analysis for the Laplace–Beltrami operator formulated on a twodimensional manifold with a fracture. We characterize the shape gradient of a functional as a bounded measure on the manifold and decompose it into a "distributed gradient" supported on the manifold, plus a singular part that we derive as the limit of a "jump" through the crack and Dirac measures at the crack extremities. The important point is that we introduce a technique that is not dimension dependent, and makes no use of classical arguments such as the maximum principle or continuation uniqueness. The technique makes use of a family of envelopes surrounding the fracture which enable us to relax certain terms and to overcome the lack of regularity resulting from the presence of the fracture. We use the min–max differentiation in order to avoid taking the derivative of the state equation and to manage the crackʹs singularities. Therefore, we write the functional in a min–max formulation on a space which takes into account the hidden boundary regularity established by the tangential extractor method.