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 Mat. Sb., 1996, Volume 187, Number 4, Pages 59–116 (Mi msb123)

Mixed problems with non-homogeneous boundary conditions in Lipschitz domains for second-order elliptic equations with a parameter

B. V. Pal'tsev

Dorodnitsyn Computing Centre of the Russian Academy of Sciences

Abstract: For a second-order elliptic equation involving a parameter, with principal part in divergence form in Lipschitz domain $\Omega$ mixed problems (of Zaremba type) with non-homogeneous boundary conditions are considered for generalized functions in $W^1_2(\Omega )$. The Poincaré–Steklov operators on Lipschitz piece $\gamma$ of the domain's boundary $\Gamma$ corresponding to homogeneous mixed boundary conditions on $\Gamma \setminus \gamma$ are studied. For a homogeneous equation with separation of variables in a tube domain with Lipschitz section, the Fourier method is substantiated for homogeneous mixed boundary conditions on the lateral surface and non-homogeneous conditions on the ends.

DOI: https://doi.org/10.4213/sm123

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English version:
Sbornik: Mathematics, 1996, 187:4, 525–580

Bibliographic databases:

UDC: 517.956
MSC: Primary 35J25; Secondary 35P10

Citation: B. V. Pal'tsev, “Mixed problems with non-homogeneous boundary conditions in Lipschitz domains for second-order elliptic equations with a parameter”, Mat. Sb., 187:4 (1996), 59–116; Sb. Math., 187:4 (1996), 525–580

Citation in format AMSBIB
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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. N. A. Meller, B. V. Pal'tsev, I. I. Chechel', “A rapidly convergent iterative domain-decomposition method for boundary-value problems for a second-order elliptic equation with a parameter”, Comput. Math. Math. Phys., 36:10 (1996), 1345–1358
2. B. V. Pal'tsev, I. I. Chechel', “Bilinear finite element implementations of iterative methods with incomplete splitting of boundary conditions for a Stokes-type system on a rectangle”, Comput. Math. Math. Phys., 39:11 (1999), 1755–1780
3. N. A. Meller, B. V. Pal'tsev, E. G. Khlyupina, “On some finite element implementations of iterative methods with splitting of boundary conditions for Stokes and Stokes-type systems in a spherical layer: Axially symmetric case”, Comput. Math. Math. Phys., 39:1 (1999), 92–117
4. V. O. Belash, B. V. Pal'tsev, “Bicubic finite-element implementations of methods with splitting of boundary conditions for a Stokes-type system in a strip under the periodicity condition”, Comput. Math. Math. Phys., 42:2 (2002), 188–210
5. Vlasov, VI, “A method for solving boundary value problems for the Laplace equation in domains with cones”, Doklady Mathematics, 70:1 (2004), 599
6. B. V. Pal'tsev, I. I. Chechel', “Second-order accurate (up to the axis of symmetry) finite-element implementations of iterative methods with splitting of boundary conditions for Stokes and stokes-type systems in a spherical layer”, Comput. Math. Math. Phys., 45:5 (2005), 816–857
7. M. S. Agranovich, “Regularity of Variational Solutions to Linear Boundary Value Problems in Lipschitz Domains”, Funct. Anal. Appl., 40:4 (2006), 313–329
8. Levitin, M, “A simple method of calculating eigenvalues and resonances in domains with infinite regular ends”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 138 (2008), 1043
9. V. I. Voititskiy, N. D. Kopachevskiy, P. A. Starkov, “Multicomponent conjugation problems and auxiliary abstract boundary-value problems”, Journal of Mathematical Sciences, 170:2 (2010), 131–172
10. M. K. Kerimov, “Boris Vasil'evich Pal'tsev (on the occasion of his seventieth birthday)”, Comput. Math. Math. Phys., 50:7 (2010), 1113–1119
11. B. V. Pal'tsev, M. B. Soloviev, I. I. Chechel', “On the development of iterative methods with boundary condition splitting for solving boundary and initial-boundary value problems for the linearized and nonlinear Navier–Stokes equations”, Comput. Math. Math. Phys., 51:1 (2011), 68–87
12. M. S. Agranovich, “Mixed Problems in a Lipschitz Domain for Strongly Elliptic Second-Order Systems”, Funct. Anal. Appl., 45:2 (2011), 81–98
13. M. S. Agranovich, “Spectral problems in Lipschitz domains”, Journal of Mathematical Sciences, 190:1 (2013), 8–33
14. B. V. Pal'tsev, “To the theory of asymptotically stable second-order accurate two-stage scheme for an inhomogeneous parabolic initial-boundary value problem”, Comput. Math. Math. Phys., 53:4 (2013), 396–430
15. N. D. Kopachevsky, “Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms”, Journal of Mathematical Sciences, 225:2 (2017), 226–264
16. 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
17. Polkovnikov A., Shlapunov A., “on Non-Coercive Mixed Problems For Parameter-Dependent Elliptic Operators”, Math. Commun., 20:2 (2015), 131–150
18. Polkovnikov A.N., “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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