Boundary integral equations on complex screens

Ralf Hiptmair (ETH)

Joint work with Xavier Claeys (UPMC)

A complex screen is an arrangement of panels that may not be even locally orientable because of junction points (2D) or edges (3D). Whereas the situation of simple screens that are locally orientable Lipschitz manifolds with boundary is well understood, the presence of junctions compounds difficulties encountered in the definition of appropriate trace spaces and boundary integral operators associated with second-order elliptic PDEs outside the screen. Our approach to overcome these difficulties is guided by the intuition that a screen is the limiting case of a massive object, heavily relies on the understanding of trace spaces as quotient spaces of functions defined on the complement of the screen, and employs Green's formula to define duality pairings in trace spaces. Using these ideas and tools, we generalize the notions of trace spaces with boundary conditions and jump traces to complex screens. In the process we introduce layer potentials and boundary integral operators for scalar second-order elliptic PDEs and derive their properties like jump relations. Extensions to electromagnetic field equations will be discussed briefly.