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The Dirichlet-to-Neumann operator via hidden compactness | | W. Arendt
; A.F.M. ter Elst
; J.B. Kennedy
; M. Sauter
; | | Date: |
3 May 2013 | | Abstract: | We show that to each symmetric elliptic operator of the form [ mathcal{A} =
- sum partial_k , a_{kl} , partial_l + c ] on a bounded Lipschitz domain
$Omega subset mathbb{R}^d$ one can associate a self-adjoint
Dirichlet-to-Neumann operator on $L_2(partial Omega)$, which may be
multi-valued if 0 is in the Dirichlet spectrum of $mathcal{A}$. To overcome
the lack of coerciveness in this case, we employ a new version of the
Lax--Milgram lemma based on an indirect ellipticity property that we call
hidden compactness. We then establish uniform resolvent convergence of a
sequence of Dirichlet-to-Neumann operators whenever their coefficients converge
uniformly and the second-order limit operator in $L_2(Omega)$ has the unique
continuation property. We also consider semigroup convergence. | | Source: | arXiv, 1305.0720 | | Services: | Forum | Review | PDF | Favorites | |
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