Layer potential operators for transmission problems on extension domains

Gabriel Claret (1, 2, 3, 4) , Michael Hinz (5) , Anna Rozanova-Pierrat (2) , Alexander Teplyaev (6)

1 CentraleSupélec
2 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes
3 Université Paris-Saclay
4 Fédération de Mathématiques de CentraleSupélec
5 Bielefeld University, Faculty of Mathematics, Bielefeld
6 UCONN - University of Connecticut

We use the well-posedness of transmission problems on classes of two-sided Sobolev extension domains to give variational definitions for (boundary) layer potential oper- ators and Neumann-Poincaré operators. These classes of domains contain Lipschitz domains, but also domains with fractal boundaries. Our formulation does not involve any measures on the boundary. We discuss basic properties of these operators and use them to generalize basic results in imaging beyond the Lipschitz case.

Keywords: Fractal boundaries Layer potentials Poincaré-Steklov operator Neumann-Poincaré operator Calderón projector elliptic transmission problems inverse problems; Fractal boundaries; Layer potentials; Poincaré-Steklov operator; Neumann-Poincaré operator; Calderón projector; elliptic transmission problems; inverse problems

Domains: Analysis of PDEs [math.AP]; Functional Analysis [math.FA]; Mathematical Physics [math-ph]

Gabriel Claret, Michael Hinz, Anna Rozanova-Pierrat, Alexander Teplyaev. Layer potential operators for transmission problems on extension domains. 2023. ⟨hal-04505158⟩