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 Mat. Tr., 2015, Volume 18, Number 2, Pages 133–204 (Mi mt297)

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. II

N. Tarkhanova, A. A. Shlapunovb

a Universität Potsdam, Institut für Mathematik, Am Neuen Palais, 10, Potsdam, 14469 GERMANY
b Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk

Abstract: We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain $\mathcal{D}$ of ${\mathbb{R}}^n$ for a second order elliptic differential operator $A (x,\partial)$. The differential operator is assumed to be of divergent form in $\mathcal{D}$ and the boundary operator $B (x,\partial)$ is of Robin type on $\partial \mathcal{D}$. The boundary of $\mathcal{D}$ is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset $Y \subset \partial \mathcal{D}$ and control the growth of solutions near $Y$. We prove that the pair $(A,B)$ induces a Fredholm operator $L$ in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set $Y$. Moreover, we prove the completeness of root functions related to $L$.

Key words: mixed problems, noncoercive boundary conditions, elliptic operators, root functions, weighted Sobolev spaces.

 Funding Agency Grant Number Alexander von Humboldt-Stiftung Russian Foundation for Basic Research 14-01-00544

DOI: https://doi.org/10.17377/mattrudy.2015.18.208

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English version:
Siberian Advances in Mathematics, 2016, 26:4, 247–293

Bibliographic databases:

Document Type: Article
UDC: 517.95+517.98

Citation: N. Tarkhanov, A. A. Shlapunov, “Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. II”, Mat. Tr., 18:2 (2015), 133–204; Siberian Adv. Math., 26:4 (2016), 247–293

Citation in format AMSBIB
\Bibitem{TarShl15} \by N.~Tarkhanov, A.~A.~Shlapunov \paper Sturm--Liouville problems in weighted spaces in domains with nonsmooth edges.~II \jour Mat. Tr. \yr 2015 \vol 18 \issue 2 \pages 133--204 \mathnet{http://mi.mathnet.ru/mt297} \crossref{https://doi.org/10.17377/mattrudy.2015.18.208} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3588295} \elib{http://elibrary.ru/item.asp?id=24639787} \transl \jour Siberian Adv. Math. \yr 2016 \vol 26 \issue 4 \pages 247--293 \crossref{https://doi.org/10.3103/S1055134416040027} 

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This publication is cited in the following articles:
1. A. N. Polkovnikov, “On the completeness of root functions of a holomorphic family of non-coercive mixed problem”, Complex Var. Elliptic Equ., 61:9 (2016), 1223–1240
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