A class of nonlinear elliptic boundary value problems

TL;DR

This paper develops an operator theoretic approach to nonlinear elliptic boundary value problems with spectral parameter-dependent boundary conditions, providing explicit solutions in special cases and generalizing known linear results.

Contribution

It introduces a linearization method for nonlinear boundary conditions and derives explicit solution operators for rational boundary functions, extending classical linear elliptic theory.

Findings

Constructed a solution operator for a broad class of nonlinear boundary conditions.

Derived explicit solutions for rational Nevanlinna or Riesz-Herglotz boundary functions.

Generalized known linear boundary value problem results to nonlinear spectral parameter cases.

Abstract

In this paper second order elliptic boundary value problems on bounded domains $\Omega\subset\dR^n$ 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.