Séminaire Analyse Numérique et EDP
Well-posedness and shape optimization on non-Lipschitz domains
jan. 2022
Intervenant : Anna Rozanova-Pierrat
Institution : Centrale Supélec
Heure : 14h00
Lieu : 3L8

We present an adaptation of the functional analysis to solve the PDEs on domains with rough/irregular boundaries. In the case of the Sobolev extension domains with a compact trace operator on its boundary it is possible to treat the weak well-posedness questions of the PDEs, not necessarily linear. The non-Lipschitz boundaries imply the absence of the H^2-regularity of the weak solutions. Still, the L^2 regularity of the Laplacian could replace it.
We present the adaptation of classical notions as the normal derivative, the trace operator. Then we give examples of the well-posed problems for elliptic and linear and non-linear wave equations with Robin type boundary conditions. This precise, in particular, results of Daners. The interest of the non-Lipschitz boundaries can be motivated by the energy minimization in the Robin type shape optimization framework. The class of Lipschitz boundaries with uniformly bounded length does not provide an energy minimum.

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