Asymptotic Analysis of Fields in Multi-Structures

by Kozlov, Vladimir; Maz'ya, Vladimir; Movchan, Alexander

• ISBN13: 9780198514954
• ISBN10: 0198514956
• Format: Hardcover
• Copyright: 1999-10-28
• Publisher: Oxford University Press

Summary

The asymptotic analysis of boundary value problems in parameter-dependent domains is a rapidly developing field of research in the theory of partial differential equations, with important applications in electrostatics, elasticity, hydrodynamics and fracture mechanics. Building on the work of Ciarlet and Destuynder, this book provides a systematic coverage of these methods in multi-structures, i.e. domains which are dependent on a small parameter e in such a way that the limit region consists of subsets of different space dimensions. An undergraduate knowledge of partial differential equations and functional analysis is assumed.

Table of Contents

List of symbols	xiv
Introduction to compound asymptotic expansions	1	(55)
Elementary examples of perturbation problems for ordinary differential equations	1	(4)
A one-dimensional singularly perturbed problem	5	(5)
Neumann boundary value problem in a domain with small cavity	10	(12)
Formulation of the problem	10	(2)
The leading order approximation	12	(1)
Remainder estimate	13	(3)
Complete asymptotic expansion	16	(4)
Asymptotic formula for the energy	20	(2)
Dirichlet boundary value problem in a domain with small inclusion	22	(8)
The leading order approximation	22	(2)
The next approximation	24	(1)
The complete asymptotic expansion	25	(5)
Mixed boundary value problem for the Laplacian in a thin rectangle	30	(8)
Formulation of the boundary value problem	30	(1)
Two-term approximation	31	(3)
The next approximation	34	(2)
Higher-order approximation	36	(2)
Problem of junction between thin bodies	38	(18)
Model problems	40	(7)
The leading order approximation	47	(2)
The next-order approximation	49	(2)
The complete asymptotic expansion	51	(2)
The remainder estimate	53	(3)
A boundary value problem for the Laplacian in a multistructure	56	(59)
Formulation of the problem	58	(1)
Model problems	59	(18)
Limit domains	59	(1)
Model problem in Ω	60	(3)
Model problem for the junction region	63	(7)
Junction layer	70	(1)
Model problem for the bottom region	71	(4)
Two model problems for a thin cylinder	75	(1)
Algebraic system for the skeleton	76	(1)
Right-hand sides	77	(4)
Local coordinates and limit domains	78	(1)
Cut-off functions	78	(1)
Asymptotic representations of the right-hand sides	79	(2)
The leading term of the asymptotic solution	81	(8)
Domain Ω	81	(2)
Junction layer	83	(2)
Thin cylinder	85	(2)
Bottom layer	87	(1)
Evaluation of W0(j)	88	(1)
Evaluation of the constants T0(j) and C0	88	(1)
Concluding remarks on formal algorithm	89	(1)
Complete asymptotic expansion	89	(3)
Structure of the asymptotic expansion	89	(2)
The asymptotic algorithm	91	(1)
Justification of the asymptotic expansion	92	(6)
Auxiliary estimates for functions in H1(ΩE)	92	(2)
Estimate for solutions	94	(3)
Estimate for the remainder term	97	(1)
A constant right-hand side	98	(2)
Application to the asymptotics of the energy integral	100	(5)
The case of the right-hand sides concentrated in Ω	101	(2)
The case of the Dirichlet data at the bases of thin cylinders	103	(2)
On a general 1D--3D multi-structure	105	(3)
A multi-structure with a thin-walled tube	108	(7)
Auxiliary facts from mathematical elasticity	115	(40)
Basic formulae of linear elasticity	115	(3)
Stress and strain	115	(1)
Equations of equilibrium and boundary conditions	116	(2)
Two-dimensional problems of linear elasticity	118	(2)
Plane strain	118	(1)
Anti-plane shear	119	(1)
Differential equations for engineering models of elastic rods	120	(1)
Classical solutions of linear elasticity for a half-space	121	(5)
Boussinesq--Cerruti's solution	121	(2)
Mindlin's solution	123	(1)
Connection between the Boussinesq--Cerruti and Mindlin solutions	124	(2)
Special solutions for a bounded two-dimensional domain	126	(3)
The torsion potential	127	(1)
The bending potentials	128	(1)
Example	128	(1)
Special solutions of linear elasticity for an infinite cylinder	129	(12)
Representation of differential operators	129	(1)
The spectral problem	130	(2)
X3-polynomial solutions	132	(1)
Biorthogonality conditions	133	(2)
The normalised stiffness coefficients	135	(1)
Biorthogonality relations for eigenvectors and generalised eigenvectors	136	(2)
There are no other polynomials	138	(3)
Green's matrix in Ω	141	(2)
Definition	141	(1)
Asymptotics	141	(2)
Korn's inequalities	143	(8)
The case of bounded Lipschitz domains	143	(3)
Half-space and a cylinder	146	(5)
Asymptotics at infinity for solutions to the traction problem for a half-cylinder	151	(4)
Elastic multi-structure	155	(58)
Multi-structure and boundary value problem	156	(2)
Model problems	158	(22)
Limit domains	158	(1)
Model problem for the body Ω	158	(2)
Junction layer	160	(7)
Model problem for the bottom layer	167	(4)
Model problem for a bounded two-dimensional domain	171	(1)
Model problems on the axis of an elastic rod	172	(2)
Model matrices and the pile structure	174	(5)
Special cases	179	(1)
Asymptotic expansion of the solution	180	(18)
Asymptotic representation of the right-hand sides	180	(2)
Description of the asymptotic series for the solution	182	(2)
Auxiliary solutions of the Lame system in a thin elastic rod	184	(3)
Expansions for displacement in a thin rod	187	(2)
Junction layer	189	(1)
Displacement in Ω	190	(1)
Bottom layer	191	(2)
Functions uk(m,j)	193	(4)
The recurrent procedure for the asymptotic expansion	197	(1)
Justification of the asymptotic expansion	198	(5)
Korn's inequality in Ωε	198	(1)
An estimate for the solution	199	(4)
The leading order approximation	203	(5)
The term u(-1)	203	(1)
The term u(0)	204	(4)
Physical interpretation of the results	208	(5)
The case |M3| + |F1| + |F2| ≠ 0	208	(2)
The case F1 = F2 = M3 = 0	210	(3)
Non-degenerate elastic multi-structures	213	(35)
Pile structure model	214	(10)
Skeleton of the multi-structure	214	(2)
The pile structure	216	(1)
Mathematical model of the pile structure	216	(1)
Solution of the pile structure equations	217	(2)
Algebraic system for the pile structure model	219	(2)
Non-degenerate and degenerate pile structures	221	(1)
Examples	222	(2)
Multi-structure and the boundary value problem	224	(3)
Description of the multi-structure	224	(2)
Formulation of the boundary value problem	226	(1)
Model problems	227	(4)
Junction layer	227	(3)
Remaining model problems	230	(1)
Asymptotic expansion of the solution	231	(13)
Cut-off functions	231	(2)
Asymptotic representation of the right-hand sides for the case of a non-degenerate multi-structure	233	(1)
Structure of the asymptotic series for the displacement field in Ωε	233	(2)
The junction layer	235	(2)
Displacement in Ω	237	(1)
The bottom layer	238	(1)
Functions ui(m,j)	239	(2)
Evaluation of the lock forces and moments at junction points	241	(1)
Algebraic system for α(m), β(m)	242	(1)
The recurrent procedure for the asymptotic expansion	243	(1)
Estimate for the remainder of the asymptotic expansion	244	(1)
Analysis of the leading term	245	(1)
Physical interpretation	246	(2)
Spectral analysis for 3D--1D multi-structures	248	(26)
An abstract scheme for the asymptotics of eigenvalues	249	(4)
Spectral problem for the Laplacian	253	(4)
The first eigenvalue	253	(2)
The first eigenfunction	255	(2)
Asymptotics of first eigenvalues of the Lame operator	257	(10)
Spectral problem	258	(1)
Korn-type inequalities	258	(5)
Spaces X0 and H0	263	(1)
Asymptotic formula for the eigenvalues	263	(4)
Spectral problem for an inhomogeneous elastic multistructure	267	(7)
The spectral problem	267	(1)
Asymptotic formulae for the eigenvalues	268	(6)
Bibliographical remarks	274	(2)
Bibliography	276	(5)
Index	281