Shape derivative of the Dirichlet energy for a transmission problem

Philippe Laurençot (1) , Christoph Walker (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IFAM - Institut für Angewandte Mathematik

For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u, the shape derivative of the Dirichlet energy (with respect to u) is shown to be well-defined and is computed. The main difficulties in this context arise from the weak regularity of the domain and the possible non-empty intersection of the graph of u and the transmission interface. The result is applied to establish the existence of a solution to a free boundary transmission problem for an electrostatic actuator.

Mots clés: obstacle problem; transmission problem; Shape derivative; free boundary problem; fourth-order equation

Domaines: Equations aux dérivées partielles [math.AP]

Philippe Laurençot, Christoph Walker. Shape derivative of the Dirichlet energy for a transmission problem. Archive for Rational Mechanics and Analysis, 2020, 237, pp.447--496. ⟨hal-01988456v2⟩