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- ItemLagrange method in shape optimization for non-linear partial differential equations: A material derivative free approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Sturm, KevinThis paper studies the relationship between the material derivative method, the shape derivative method, the min-max formulation of Correa and Seeger, and the Lagrange method introduced by Céa. A theorem is formulated which allows a rigorous proof of the shape differentiability without the usage of material derivative; the domain expression is automatically obtained and the boundary expression is easy to derive. Furthermore, the theorem is applied to a cost function which depends on a quasi-linear transmission problem. Using a Gagliardo penalization the existence of optimal shapes is established.