Geodesic
Existence criteria in some extremum problems involving eigenvalues of elliptic operators
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 9 (2016) no. 1, pp. 37-47.

Voir la notice de l'article provenant de la source Math-Net.Ru

Some existence criteria for a certrain class of extremum problems involving eigenvalues of linear elliptic boundary-value problems (including ones in the form of variational inequalities) are proved. The approach applied admits an extension to the case of extremum problems associated with eigenvalues of nonlinear boundary-value problems. Some applications to optimal structural design and comparisons with results in the literature are given.
Keywords: eigenvalue optimization problem, elliptic boundary-value problem, variational inequality, existence theorem, optimal structural design. 
@article{JSFU_2016_9_1_a4,
     author = {Vasily Yu. Goncharov},
     title = {Existence criteria in some extremum problems involving eigenvalues of elliptic operators},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {37--47},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2016_9_1_a4/}
}
TY  - JOUR
AU  - Vasily Yu. Goncharov
TI  - Existence criteria in some extremum problems involving eigenvalues of elliptic operators
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2016
SP  - 37
EP  - 47
VL  - 9
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2016_9_1_a4/
LA  - en
ID  - JSFU_2016_9_1_a4
ER  - 
%0 Journal Article
%A Vasily Yu. Goncharov
%T Existence criteria in some extremum problems involving eigenvalues of elliptic operators
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2016
%P 37-47
%V 9
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2016_9_1_a4/
%G en
%F JSFU_2016_9_1_a4
Vasily Yu. Goncharov. Existence criteria in some extremum problems involving eigenvalues of elliptic operators. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 9 (2016) no. 1, pp. 37-47. http://geodesic.mathdoc.fr/item/JSFU_2016_9_1_a4/

[1] N. V. Banichuk, Problems and Methods of Optimal Structural Design, Plenum Press, 1983 | MR

[2] W. G. Litvinov, Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics, Birkhäuser Verlag, 2000 | MR | Zbl

[3] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser Verlag, 2006 | MR | Zbl

[4] I. Bock, “On optimal control problems connected with eigenvalue variational inequalities”, Math. Meth. Appl. Sci., 24:9 (2001), 671–688 | DOI | MR | Zbl

[5] B. Schwarz, “Some results on the frequencies of nonhomogeneous rods”, J. Math. Anal. Appl., 5:2 (1962), 169–175 | DOI | MR | Zbl

[6] S. J. Cox, J. R. McLaughlin, “Extremal eigenvalue problems for composite membranes, I”, Appl. Math. Optim., 22:1 (1990), 153–167 | DOI | MR | Zbl

[7] S. J. Cox, M. L. Overton, “On the optimal design of columns against buckling”, SIAM J. Math. Anal., 23:2 (1992), 287–325 | DOI | MR | Zbl

[8] Y. V. Egorov, V. A. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser Verlag, 1996 | MR | Zbl

[9] A. Myslinsky, J. Sokolowski, “Nondifferentiable optimization problems for elliptic systems”, SIAM J. Control and Optimization, 23:4 (1985), 632–648 | DOI | MR

[10] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, 2011 | MR | Zbl

[11] E. Miersemann, “Verzweigungsprobleme für variationsungleichugen”, Math. Nach., 65:1 (1975), 187–209 | DOI | MR | Zbl

[12] B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals, Lecture Notes in Mathematics, 922, Springer-Verlag, 1982 | MR

[13] B. T. Polyak, “Semicontinuity of integral functionals and existence theorems on extremal problems”, Math. USSR Sb., 7:1 (1969), 59–77 | DOI

[14] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980 | Zbl