A class of nonlinear elliptic boundary value problems

Abstract

In this paper second order elliptic boundary value problems on bounded domains $Ω\subset\dR^n$ with boundary conditions on $\partialΩ$ 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 $L^2(Ω)\oplus (L^2(\partialΩ))^m$, 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.