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A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains
, 1995
"... Introduction We study continuity of boundary problems with varying domains. To explain this in more detail, let us consider our standard example: Denote by HGn the Dirichlet Laplacian on the open set G n ae IR d . The basic question which we adress is, whether we have convergence HGn \Gamma! H ..."
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Cited by 10 (3 self)
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Introduction We study continuity of boundary problems with varying domains. To explain this in more detail, let us consider our standard example: Denote by HGn the Dirichlet Laplacian on the open set G n ae IR d . The basic question which we adress is, whether we have convergence HGn \Gamma! HG ; if the sets G n converge to G in an appropriate sense. Two notions of convergence for the operators appear suitable: Generalized convergence in the strong resolvent sense (srs) and in the norm resolvent sense (nrs) (the "generalized" refers to the fact that the HGn act in different Hilbert spaces; we will frequently omit it). We shall introduce these concepts in some detail below but first we briefly describe the content of the following sections. In Section 1 we a
Sch’nol’s theorem for strongly local forms
, 2009
"... We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions. ..."
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Cited by 10 (6 self)
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We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions.
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 6 (5 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 4 (4 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.