Séminaire Anna Rozanova-Pierrat (École Centrale Supelec)
Based on my recent publications, I will present a generalization 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.
I will explain the adaptation of classical notions as the normal derivative and the trace operator. Then I will give examples of the well-posed problems for elliptic and linear and non-linear wave equations with Robin type boundary conditions (at least on a non trivial part of the boundary). This precise, in particular, results of Daners. The main examples are the linear wave equation and its frequency version the Helmholtz equation, the strong damped wave equation and also the Westervelt equation.
The Helmholtz equation with the Robin type boundary conditions with a complex parameter models well the partial absorption of an acoustical wave by a boundary, viewed as an interface between the air and a porous medium. The Westervelt equation (a non linear version of the strongly damped wave equation useful in the ultrasound propagation) has a dissipative term describing an absorption in the medium and the boundary reflection could be modelled with the real-valued Robin boundary condition.
In the end of my talk I will briefly present the results related with the Mosco convergence. Established Mosco convergence of energies and weak formulation functionals for a converging sequence of domains (typically, prefractal to fractal) provides an approximation of a solution (of the Helmholtz or Westervelt problems) on a domain with irregular boundary (for example, fractal). 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 not necessarily provides an energy minimum. However the existence of an optimal shape is guaranteed in a larger class allowing non-Lipschitz domains. Related applications of this theory for the Helmholtz problem could be optimization of anti-noise walls or anechoic chambers.
The results are obtained in the collaboration with A. Dekkers, M. Hinz, F. Magoulès, P. Omnès and A. Teplyaev.