Anna Rozanova-Pierrat (Centrale Supélec)

Boundary value problems on domains with non-Lipschitz boundaries and applications in the shape optimisation.

We present main functional analysis aspects allowing to solve the PDEs on domains with rough/irregular/fractal/non-Lipschitz 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 H2-regularity of the weak elliptic solutions. Still, the L2 regularity of the Laplacian could replace it. We give examples of the well-posed problems for elliptic and linear and non-linear wave equations with Robin type boundary conditions. The interest of the non-Lipschitz boundaries can be motivated by the energy minimization in the Robin type shape optimization framework. We start by proving the Mosco convergence of the functionals corresponding to the energy/variational formulations on a sequence of domains and the convergence of the spectrum (which does not follows from the Mosco convergence). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall ensures the infimum of the acoustic energy. Then we also introduce a larger compact class of (ε, ∞) - or uniform domains with possibly non-Lipschitz (for ex- ample, fractal) boundaries on which an optimal shape exists, giving the minimum of the energy. The boundaries are described as the supports of Radon measures ensuring their Hausdorff dimension in the segment [n−1, n) . For a fixed Lipschitz or non-Lipschitz boundary, we also solve theoretically and numerically a parametric shape optimization problem to find the optimal distribution of absorbing material in the reflexive one minimizing the acoustical energy on a frequency range.