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 $\partial \Omega$ 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 $L^2(\Omega )\oplus (L^2(\partial \Omega ){)}^m$, which is a natural generalization of known results on $\lambda$-linear elliptic boundary value problems and $\lambda$-rational boundary value problems for ordinary second order differential equations.

References in corpus (5)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Krein's Resolvent Formula for Self-Adjoint Extensions of Symmetric Second Order Elliptic Differential Operators
• Krein resolvent formulas for elliptic boundary problems in nonsmooth domains
• First order operators and boundary triples
• Spectral Theory of Elliptic Operators in Exterior Domains