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 J. Sib. Fed. Univ. Math. Phys., 2013, Volume 6, Issue 2, Pages 247–261 (Mi jsfu311)

On the spectral properties of a non-coercive mixed problem associated with $\overline\partial$-operator

Alexander N. Polkovnikov, Aleksander A. Shlapunov

Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk, Russia

Abstract: We consider a non-coercive Sturm–Liouville boundary value problem in a bounded domain $D$ of the complex space $\mathbb C^n$ for the perturbed Laplace operator. More precisely, the boundary conditions are of Robin type on $\partial D$ while the first order term of the boundary operator is the complex normal derivative. We prove that the problem is Fredholm one in proper spaces for which an Embedding Theorem is obtained; the theorem gives a correlation with the Sobolev–Slobodetskii spaces. Then, applying the method of weak perturbations of compact self-adjoint operators, we show the completeness of the root functions related to the boundary value problem in the Lebesgue space. For the ball, we present the corresponding eigenvectors as the product of the Bessel functions and the spherical harmonics.

Keywords: Sturm–Liouville problem, non-coercive problems, the multidimensional Cauchy–Riemann operator, root functions.

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UDC: 517.95+517.5
Received in revised form: 10.01.2013
Accepted: 20.01.2013
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Citation: Alexander N. Polkovnikov, Aleksander A. Shlapunov, “On the spectral properties of a non-coercive mixed problem associated with $\overline\partial$-operator”, J. Sib. Fed. Univ. Math. Phys., 6:2 (2013), 247–261

Citation in format AMSBIB
\Bibitem{PolShl13} \by Alexander~N.~Polkovnikov, Aleksander~A.~Shlapunov \paper On the spectral properties of a~non-coercive mixed problem associated with $\overline\partial$-operator \jour J. Sib. Fed. Univ. Math. Phys. \yr 2013 \vol 6 \issue 2 \pages 247--261 \mathnet{http://mi.mathnet.ru/jsfu311} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Shlapunov A., Tarkhanov N., “On Completeness of Root Functions of Sturm-Liouville Problems with Discontinuous Boundary Operators”, J. Differ. Equ., 255:10 (2013), 3305–3337
2. N. Tarkhanov, A. A. Shlapunov, “Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. II”, Siberian Adv. Math., 26:4 (2016), 247–293
3. Polkovnikov A., Shlapunov A., “on Non-Coercive Mixed Problems For Parameter-Dependent Elliptic Operators”, Math. Commun., 20:2 (2015), 131–150
4. Shlapunov A., Peicheva A., “on the Completeness of Root Functions of Sturm-Liouville Problems For the Lame System in Weighted Spaces”, ZAMM-Z. Angew. Math. Mech., 95:11 (2015), 1202–1214
5. Anastasiya S. Peicheva, “Embedding theorems for functional spaces associated with a class of Hermitian forms”, Zhurn. SFU. Ser. Matem. i fiz., 10:1 (2017), 83–95
6. A. N. Polkovnikov, A. A. Shlapunov, “Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions”, Siberian Math. J., 58:4 (2017), 676–686
7. A. Laptev, A. Peicheva, A. Shlapunov, “Finding eigenvalues and eigenfunctions of the Zaremba problem for the circle”, Complex Anal. Oper. Theory, 11:4 (2017), 895–926
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