Junctions of singularly degenerating domains with different limit dimensions. II

This paper is aimed at finding asymptotic formulas for solutions to the mixed boundary problem for the Poisson equation in a domain obtained by joining singularly degenerating domains. In this paper, which is the second part of the work (the first part was published in No. 18), the main attention is given to three-dimensional problems in which a thin plate or a periodic family of thin rods is joined to a massive body (the distances between the rods are comparable with the diameters of their cross-sections). The distinctive feature of such problems is that an integral equation arises as one of the limit problems. Bibliography: 48 titles.

[1]  I. Argatov,et al.  Junction problem of shashlik (skewer) type , 1993 .

[2]  C. Anné.  Spectre du laplacien et écrasement d'anses , 1987 .

[3]  V. Gorbachuk,et al.  Boundary Value Problems for Operator Differential Equations , 1990 .

[4]  D. Leguillon,et al.  Approximation of a two dimensional problem of junctions , 1990 .

[5]  O. Oleinik,et al.  Boundary-value problems for partial differential equations in non-smooth domains , 1983 .

[6]  S. Ozawa,et al.  Singular Hadamard's variation of domains and eigenvalues of the Laplacian, II , 1980 .

[7]  G. Pólya,et al.  Isoperimetric inequalities in mathematical physics , 1951 .

[8]  Yu. V. Pleskov,et al.  Russian Text Ignored , 1962 .

[9]  Philippe G. Ciarlet,et al.  Plates And Junctions In Elastic Multi-Structures , 1990 .

[10]  M. Vishik,et al.  ELLIPTIC PROBLEMS WITH A PARAMETER AND PARABOLIC PROBLEMS OF GENERAL TYPE , 1964 .

[11]  O. A. Ladyzhenskai︠a︡.  Boundary value problems of mathematical physics , 1967 .

[12]  O. Oleinik,et al.  Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities , 1988 .

[13]  M. Aufranc.  Numerical study of a junction between a three-dimensional elastic structure and a plate , 1989 .

[14]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[15]  N. S. Landkof.  Foundations of Modern Potential Theory , 1972 .

[16]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[17]  V. A. Kondrat'ev,et al.  Boundary problems for elliptic equations in domains with conical or angular points , 1967 .

[18]  O. Oleinik,et al.  On the asymptotic behaviour at infinity of solutions in linear elasticity , 1982 .

[19]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[20]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[21]  S. Nazarov,et al.  Edge effect in the bending of a thin three-dimensional plate , 1989 .

[22]  A. Kochubei,et al.  Extension theory for symmetric operators and boundary value problems for differential equations , 1989 .

[23]  Ja. A. Roitberg,et al.  a Theorem on Homeomorphisms for Elliptic Systems and its Applications , 1969 .

[24]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .

[25]  Asymptotic expansions of solutions of ordinary differential equations in Hilbert space , 1967 .

[26]  A. L. Goldenveizer.  THEORY OF ELASTIC THIN SHELLS , 1962 .

[27]  O. Ladyzhenskaya.  The Boundary Value Problems of Mathematical Physics , 1985 .

[28]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[29]  B. Mampassi.  Un type de jonction bidimensionnelle d'une tige et d'un massif élastiques , 1992 .

[30]  G. Polya,et al.  Isoperimetric Inequalities in Mathematical Physics. (AM-27), Volume 27 , 1951 .

[31]  B. Plamenevskii,et al.  Elliptic Problems in Domains with Piecewise Smooth Boundaries , 1994 .

[32]  Frederic Y. M. Wan,et al.  Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory , 1984 .

[33]  L. Hörmander.  Linear Partial Differential Operators , 1963 .

[34]  S. Lang,et al.  Partial Differential Operators , 1985 .

[35]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .

[36]  S. Nazarov.  ELLIPTIC BOUNDARY VALUE PROBLEMS WITH PERIODIC COEFFICIENTS IN A CYLINDER , 1982 .