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6 - Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples

Published online by Cambridge University Press:  05 November 2012

By
J. Behrndt  and
M. Langer
Edited by
Seppo Hassi ,
Hendrik S. V. de Snoo  and
Franciszek Hugon Szafraniec
J. Behrndt
Affiliation:
Technische Universität Graz
M. Langer
Affiliation:
University of Strathclyde
Seppo Hassi
Affiliation:
University of Vaasa, Finland
Hendrik S. V. de Snoo
Affiliation:
Rijksuniversiteit Groningen, The Netherlands
Franciszek Hugon Szafraniec
Affiliation:
Jagiellonian University, Krakow
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Summary

Abstract The notion of quasi boundary triples and their Weyl functions is reviewed and applied to self-adjointness and spectral problems for a class of elliptic, formally symmetric, second order partial differential expressions with variable coefficients on bounded domains.

Introduction

Boundary triples and associated Weyl functions are a powerful and ef- fficient tool to parameterize the self-adjoint extensions of a symmetric operator and to describe their spectral properties. There are numerous papers applying boundary triple techniques to spectral problems for various types of ordinary differential operators in Hilbert spaces; see, e.g. [Behrndt and Langer, 2010; Behrndt, Malamud and Neidhardt, 2008; Behrndt and Trunk, 2007; Brasche, Malamud and Neidhardt, 2002; Brüning, Geyler and Pankrashkin, 2008; Derkach, Hassi and de Snoo, 2003; Gorbachuk and Gorbachuk, 1991; Derkach and Malamud, 1995; Karabash, Kostenko and Malamud, 2009; Kostenko and Malamud, 2010; Posilicano, 2008] and the references therein.

The abstract notion of boundary triples and Weyl functions is strongly inspired by Sturm-Liouville operators on a half-line and their Titchmarsh -Weyl coefficients. To make this more precise, let us consider the ordinary differential expression l = −D2 + q on the positive half-line ℝ+ = (0, ∞), where D denotes the derivative, and suppose that q is a real-valued L∞-function. The maximal operator associated with l in L2(ℝ+) is defined on the Sobolev space H2(ℝ+) and turns out to be the adjoint of the minimal operator S f = l(f), dom S =, where is the subspace of H2(ℝ+) consisting of functions f that satisfy the boundary conditions f(0) = f′(0) = 0.

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Operator Methods for Boundary Value Problems , pp. 121 - 160
Publisher: Cambridge University Press
Print publication year: 2012

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