×

Interaction of concentrated masses in a harmonically oscillating spatial body with Neumann boundary conditions. (English) Zbl 0791.35090

The author considers a Neumann eigenvalue problem of the type \[ \Delta u+ \lambda\gamma (\varepsilon,x) u=0 \quad\text{in } \Omega\subset \mathbb{R}^ 3, \qquad \partial u/\partial\gamma= 0 \quad \text{on }\partial\Omega. \] Here the “weight-function” \(\gamma(\varepsilon,x)\) describes masses, concentrated in some points of \(\Omega\), as \(\varepsilon\to 0\). The asymptotic behaviour of this eigenvalue problem is then investigated.
Reviewer: R.Sperb (Zürich)

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] E. SANCHEZ-PALENCIA, 1984, Perturbation of Eigenvalues in Thermoelasticity and Vibration of Systems with Concentrated Masses, Lecture Notes in Physics,195, Berlin, Heidelberg, New York : Springer, 346-368. Zbl0542.73006 MR755735 · Zbl 0542.73006
[2] E. SANCHEZ-PALENCIA, H. TCHTAT, 1984, Vibration de systèmes élastiques avec masses concentrées, Rend. Sem. Mat. Univ. Politec. Torino, 42, 43-63. Zbl0658.73044 MR834781 · Zbl 0658.73044
[3] C. LEAL, J. SANCHEZ-HUBERT, 1989, Perturbation of the eigenvalues of a membrane with concentrated mass. Quart. Appl. Math., vol. 47, 93-103. Zbl0685.73025 MR987898 · Zbl 0685.73025
[4] U. A. GOLOVATII, S. A. NAZAROV, O. A. OLEINIK, 1990, Asymptotic decompositions of eigenvalues and eigenfunctins of problems on oscillating media with concentrated masses, Trudy Mat. inst. A.N S.S.S.R., 192, 42-60. Zbl0728.35077 MR1097888 · Zbl 0728.35077
[5] J. SANCHEZ-HUBERT, E. SANCHEZ-PALENCIA, 1989, Vibration and Coupling of Continuous Systems Asymptotic Methods, Berlin, Heidelberg, New York, London, Paris, Tokyo : Springer-Verlag. Zbl0698.70003 MR996423 · Zbl 0698.70003
[6] O. A. OLEINIK, G. A. YOSIFIAN, A. S. SHAMAEV, 1990, Mathematical Problems in Theory of Non-Homogeneous Media, Moscow : Izdat. Moskov. Universiteta. · Zbl 0572.73059
[7] V. G. MAZ’YA, S. A. NAZAROV, B. A. PLAMENEVSKII, 1981, On the asymptotics of solutions of elliptic boundary value problems in domains perturbed irregularly, Probl. mat. anal., 8, Leningrad : izdat. Leningrad Universiteta, 72-153 (Russian). Zbl0491.35013 MR658154 · Zbl 0491.35013
[8] W. G. MAZJA, S. A. NASAROW, B. A. PLAMENEWSKI, 1990, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, Bd. 1, Berlin : Akademie-Verlag.
[9] V. G. MAZ’YA, S. A. NAZAROV, B. A. PLAMEENVSKII, 1983, On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone, Matem. Sbornik, 122, 435-436 (Russian ; English transl. (1987) in Math. USSR Sbornik, 57, 317-349). Zbl0599.35056 · Zbl 0599.35056 · doi:10.1070/SM1985v050n02ABEH002837
[10] S. A. NAZAROV, 1986, Justification of asymptotic expansions of the eigenvalues of nonselfadjoint singularly perturbed elliptic boundary value problems, Matem.sbornik, 129, 307-337 (Russian ; English transl. (1987) in Math. USSR Sbornik,57, 317-349). Zbl0618.35005 MR837128 · Zbl 0618.35005 · doi:10.1070/SM1987v057n02ABEH003071
[11] V. G. MAZ’YA, S. A. NAZAROV, 1989, On the singularities of solutions of the Neumann problem at a conical point, Sibirsk. Matem. Zh., 30, 52-63 (Russian). Zbl0701.35021 MR1010835 · Zbl 0701.35021 · doi:10.1007/BF00971492
[12] V. A. KONDRAT’EV, 1967, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Mat. Obshch., 16, 209-292 (Russian ; English transl. (1967) in Trans. Moscow Math. Soc., 16). Zbl0162.16301 MR226187 · Zbl 0162.16301
[13] S. A. NAZAROV, B. A. PLAMENEVSKII, 1991, Elliptic Problems in Domainswith Piecewise Smooth Boundaries, Moscow : Nauka (Russian).
[14] S. A. NAZAROV, 1989, On the Sanchez-Palencia problem with the Neumann boundary conditions, Izvestija VUZ. Matem. No. 11, 60-66 (Russian). Zbl0801.35092 MR1045104 · Zbl 0801.35092
[15] I. C. GOGBERG, M. G. KREIN, 1965, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, Moscow : Nauka (Russian ; English transl. (1969). Amer. Math. Soc., Providence, R.I.). Zbl0181.13504 · Zbl 0181.13503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.