Boundary-value problems for partial differential equations in non-smooth domains

Кондратьев, В А; Oleinik, O A

doi:10.1070/rm1983v038n02abeh003470

Cited by 187 publications

(107 citation statements)

References 78 publications

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“…Mention that the classical paper [Slo58] was actually motivated by the first boundary problem for the heat equation in a bounded domain G ⊂ R n . On the other hand, [Kon66] made essential use of function spaces of Slobodetskii [Slo58]. Unfortunately, [Kon66] suffers several drawbacks which, however, do not affect the main result of this seminal paper.…”

Section: Introductionmentioning

confidence: 99%

“…A more careful analysis led Petrovskii in [Pet34] to an explicit necessary and sufficient condition for a boundary point to be regular. This latter paper initiated an extensive literature devoted to general boundary value problems for parabolic equations, see [Mik63], [Kon66], etc. Mention that the classical paper [Slo58] was actually motivated by the first boundary problem for the heat equation in a bounded domain G ⊂ R n .…”

Section: Introductionmentioning

confidence: 99%

“…At the end of the '90s Kondrat'ev called the last author's attention to the paper [Kon66] saying "Here are cusps." In spite of the fact that [Kon66] deals with C ∞ boundaries the analysis near characteristic boundary points reveals Fuchs-type operators typical for conical singularities, provided that the contact degree of the boundary and characteristic plane is at least the anisotropy quotient (2 for the heat equation). If the contact degree is less than the anisotropy quotient, the analysis close to the characteristic point requires pseudodifferential operators typical for cuspidal points on the boundary, cf.…”

Section: Introductionmentioning

confidence: 99%

“…According to the MathSciNet of the AMS there has been merely 8 citations to the paper [Kon66] while this latter already contains all of the techniques of [Kon67], especially the asymptotics of solutions at conical points. At the end of the '90s Kondrat'ev called the last author's attention to the paper [Kon66] saying "Here are cusps."…”

Section: Introductionmentioning

confidence: 99%

“…As filtration Kondrat'ev used in [Kon66] weighted Slobodetskii spaces, where the weight functions are powers of the distance to the characteristic point. Analysis on manifolds with point and more general singularities has since exploited weighted function spaces.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

2015

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Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Abstract. The Dirichlet problem for the heat equation in a bounded domain G ⊂ R n+1 is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character. Universitätsverlag

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

2015

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Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1

Mel'nyk

2000

Math. Meth. Appl. Sci.

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Plates and Shells: Asymptotic Expansions and Hierarchic Models

Dauge

Faou

Yosibash

2004

Encyclopedia of Computational Mechanics

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Concerning thin structures such as plates and shells, the idea of reducing the equations of elasticity to twodimensional models defined on the mid-surface seems relevant. Such a reduction was first performed thanks to kinematical hypotheses about the transformation of normal lines to the mid-surface. As nowadays, the asymptotic expansion of the displacement solution of the three-dimensional linear model is fully known at least for plates and clamped elliptic shells, we start from a description of these expansions in order to introduce the twodimensional models known as hierarchical models: These models extend the classical models, and pre-suppose the displacement to be polynomial in the thickness variable, transverse to the mid-surface. Because of the singularly perturbed character of the elasticity problem as the thickness approaches zero, boundary-or internal layers may appear in the displacements and stresses, and so may numerical locking effects. The use of hierarchical models, discretized by higher degree polynomials (p-version of finite elements) may help to overcome these severe difficulties.

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Boundary-value problems for partial differential equations in non-smooth domains

Cited by 187 publications

References 78 publications

Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1

Plates and Shells: Asymptotic Expansions and Hierarchic Models

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