The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

Introduction.- Operators on Networks.- Function Spaces on Networks.- Operator Semigroups.- And Now Something Completely Different: A Crash Course in Cortical Modeling.- Sequilinear Froms and Analytic Semigroups.- Evolution Equations Associated With Self-adjoint Operators.- Symmetry Properties.- Index.

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Semigroup Methods for Evolution Equations on Networks

On the scientific work of M. Sh. Birman in 1998-2007 by M. Solomyak and T. Suslina Continuation of the list of publications of M. Sh. Birman by T. Suslina and D. Yafaev Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions by M. Sh. Birman Asymptotic behavior of the eigenfunctions of many-particle Schrodinger operator. I. One-dimensional particles by V. S. Buslaev and S. B. Levin Spectral asymptotics of the Maxwell operator on Lipschitz manifolds with boundary by M. N. Demchenko and N. D. Filonov Spectral inequalities for Schrodinger operators with surface potentials by R. L. Frank and A. Laptev On the spectrum of the Dirichlet Laplacian in a narrow infinite strip by L. Friedlander and M. Solomyak Hardy inequalities for simply connected planar domains by A. Laptev and A. V. Sobolev Lyapunov functions of periodic matrix-valued Jacobi operators by E. Korotyaev and A. Kutsenko The spectral flow, the Fredholm index, and the spectral shift function by A. Pushnitski On the spectrum of a translationally invariant Pauli operator by G. Raikov Discrete spectrum distribution of the Landau operator perturbed by an expanding electric potential by G. Rozenblum and A. V. Sobolev. On the comparison of the Dirichlet and Neumann counting functions by Y. Safarov Absolutely continuous spectrum of multi-dimensional Schrodinger operators with slowly decaying potentials by O. Safronov On discrete spectrum of the perturbed periodic magnetic Schrodinger operator with degenerate lower edge of the spectrum by R. Shterenberg Homogenization of periodic second order differential operators including first order terms by T. A. Suslina Improved Berezin-Li-Yau inequalities with a remainder term by T. Weidl Spectral and scattering theory of fourth order differential operators by D. R. Yafaev.

Spectral Theory of Differential Operators: M. Sh. Birman 80th Anniversary Collection

Matching of asymptotic expansions of solutions of boundary value problems

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Junction of a periodic family of elastic rods with a 3d plate. Part I

A convergence theorem and asymptotic estimates as C 0 are proved for a solution to a mixed boundary-value problem for the Poisson equation in a junction Q, of a domain o and a large number N 2 of c-periodically situated thin cylinders with thickness of order e = °(*) For this junction, we construct an extension operator and study its properties.

https://www.ems-ph.org/journals/show_pdf.php?issn=0232-2064&vol=18&iss=4&rank=9

Homogenization of the Poisson Equation in a Thick Periodic Junction

We investigate the asymptotic behavior, as $\varepsilon$ tends to $0^+$, of the transverse displacement of a Kirchhoff–Love plate composed of two domains $\Omega_\varepsilon^+\cup\Omega^-_\varepsil...

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Boundary Homogenization and Reduction of Dimension in a Kirchhoff–Love Plate

We consider a boundary-value problem for the Poisson equation in a thick junction Ωe, which is the union of a domain Ω0 and a large number of e-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂νue + eκ(ue)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as e  0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as e  0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H1(Ωe) is proved. Copyright © 2007 John Wiley & Sons, Ltd.

Homogenization of a boundary‐value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

Nous considerons les valeurs propres λ 0 (e) < λ 1 (e) ≤ ... du probleme de Neumann pour l'operateur de Laplace dans le domaine Ω e obtenu en ajoutant des cylindres minces sur la surface laterale d'une plaque mince (Ω e apparait comme un peigne). Le spectre du probleme limite bidimensionnel correspondant contient des valeurs propres ordinaires ainsi que des points d'accumulation P m+1 (m = 0, 1, ...) qui divisent les valeurs propres en des suites infinies {μ k m } ⊂ (P m , P m+1 ). Pour chaque m, k ≥ 0 il existe une suite {λ n(e,k,m) (e)} qui converge vers μ k m lorsque e → +0

Asymptotic structure of the spectrum of the Neumann problem in a thin comb-like domain

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ωe, which is the union of a domain Ω0 and a large number 2N of thin rods with variable thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are e-periodically alternated. The Robin conditions are given on the lateral boundaries of the thin rods. Using the method of matched asymptotic expansions, we construct the asymptotic approximation for the solution as e → 0 and prove the corresponding estimates in the Sobolev space H1(Ωe).

Taras A. Mel'nyk

Papers

Homogenization of the Poisson Equation in a Thick Periodic Junction

Boundary Homogenization and Reduction of Dimension in a Kirchhoff–Love Plate

Homogenization of a boundary‐value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

Asymptotic structure of the spectrum of the Neumann problem in a thin comb-like domain

Asymptotic approximation for the solution to the robin problem in a thick multi-level junction