Elliptic boundary value problems with λ-dependent boundary conditions

Abstract

In this paper second order elliptic boundary value problems on bounded domains Ω ⊂ R n with boundary conditions on ∂ Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz–Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L 2 ( Ω ) ⊕ ( L 2 ( ∂ Ω ) ) m , which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.