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

This article is cited in 18 scientific papers (total in 18 papers)

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
Received: 10.01.1995 and 08.09.1995

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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\by B.~V.~Pal'tsev
\paper Mixed problems with non-homogeneous boundary conditions in Lipschitz domains for second-order elliptic equations with a~parameter
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\yr 1996
\vol 187
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\pages 59--116
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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  mathscinet  zmath  isi
    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  mathnet  mathscinet  zmath  elib
    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  mathnet  mathscinet  zmath  elib
    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  mathnet  mathscinet  zmath  elib
    5. Vlasov, VI, “A method for solving boundary value problems for the Laplace equation in domains with cones”, Doklady Mathematics, 70:1 (2004), 599  mathscinet  zmath  isi  elib  scopus
    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  mathnet  mathscinet  zmath  elib
    7. M. S. Agranovich, “Regularity of Variational Solutions to Linear Boundary Value Problems in Lipschitz Domains”, Funct. Anal. Appl., 40:4 (2006), 313–329  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  mathscinet
    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  mathnet  crossref  mathscinet  adsnasa  isi  elib
    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  mathnet  crossref  mathscinet  isi  elib
    12. M. S. Agranovich, “Mixed Problems in a Lipschitz Domain for Strongly Elliptic Second-Order Systems”, Funct. Anal. Appl., 45:2 (2011), 81–98  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. M. S. Agranovich, “Spectral problems in Lipschitz domains”, Journal of Mathematical Sciences, 190:1 (2013), 8–33  mathnet  crossref  mathscinet
    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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    15. N. D. Kopachevsky, “Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms”, Journal of Mathematical Sciences, 225:2 (2017), 226–264  mathnet  crossref
    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  mathnet  crossref  crossref  mathscinet  elib
    17. Polkovnikov A., Shlapunov A., “on Non-Coercive Mixed Problems For Parameter-Dependent Elliptic Operators”, Math. Commun., 20:2 (2015), 131–150  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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