We investigate an optimization problem (OP) in a non-standard form: the cost functional F measures the L 1 distance between the solution u ϕ of the direct Robin problem and a function f ∈ L 1 (M). After proving positivity, monotonicity and control properties of the state u ϕ with respect to ϕ, we prove the existence of an optimal control ψ to the problem (OP) and establish Newton differentiability of the functional F. As an application to this optimization problem the inverse problem of determining a Robin parameter ϕ inv by measuring the data f on M is considered. In that case f is assumed to be the trace on M of u ϕ inv . In spite of the fact that we work with the L 1 -norm we prove differentiability of the cost functional F by using complex analysis techniques. The proof is strongly related to positivity and monotonicity of the derivative of the state with respect to ϕ. An identifiability result is also proved for the set of admissible parameters ad consisting of positive functions in L ∞ .

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We are interested in the study of quasistatic visco-plastic flows with thermal effects. The fluid motion is governed by the incompressible Norton-Hoff model coupled with the time-dependent heat equation where the dissipated mechanical power is the source term. The viscosity of the fluid is modeled by the non-linear Arrhenius law. The well-posedness of each decoupled system is given. The optimal regularities of the heat solution and of the scale factor are supplied. A non-linear operator describing the stand coupling is provided. The existence of a solution to the considered problem is established. We prove the compactness result of the set solutions.

This paper considers shape sensitivity analysis for the Laplace-Beltrami operator formulated on a two-dimensional 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 crack's 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. r