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2003 Wentzell boundary conditions in the context of Dirichlet forms
Hendrik Vogt, Jürgen Voigt
Adv. Differential Equations 8(7): 821-842 (2003). DOI: 10.57262/ade/1355926813

Abstract

Let $X$ be a locally compact space, $m$ a Radon measure on $X$,\, ${{\mathfrak h}}$ a regular Dirichlet form in $L_2(X,m)$. For a Radon measure $\mu$ we interpret ${{\mathfrak h}}$ as a regular Dirichlet form $\tau$ in $L_2(m+\mu)$. We show that $\mu$ decomposes as $\mu_r+\mu_s$, where $\mu_r$ is coupled to ${{\mathfrak h}}$ and $\mu_s$ decouples from ${{\mathfrak h}}$. Additionally to this `space perturbation', a second perturbation is introduced by a measure $\nu$ describing absorption. The main object of the paper is to apply this setting to a study of the Wentzell boundary condition \[ -\alpha Au + n\cdot a\nabla u + \gamma u = 0 \quad \text{on}\ \partial\Omega \] for an elliptic operator $A=-\nabla{\hspace{-0.1em}}\cdot(a\nabla)$, where $\Omega\subseteq{\mathbb{R}}^d$ is open, $n$ the outward normal, and $\alpha$, $\gamma$ are suitable functions. It turns out that the previous setting can be applied with $\mu=\alpha dS$, $\nu=\gamma dS$, under suitable conditions. Besides the description of the $d$-dimensional case we give a more detailed analysis of the one-dimensional case. As a further topic in the general setting we study the question whether mass conservation carries over from the unperturbed form ${{\mathfrak h}}$ to the space perturbed form $\tau$. In the Appendix we extend a known closability criterion from the minimal to the maximal form.

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Hendrik Vogt. Jürgen Voigt. "Wentzell boundary conditions in the context of Dirichlet forms." Adv. Differential Equations 8 (7) 821 - 842, 2003. https://doi.org/10.57262/ade/1355926813

Information

Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1040.47032
MathSciNet: MR1988680
Digital Object Identifier: 10.57262/ade/1355926813

Subjects:
Primary: 31C25
Secondary: 34B05 , 35J25 , 47B25 , 47N20

Rights: Copyright © 2003 Khayyam Publishing, Inc.

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Vol.8 • No. 7 • 2003
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