On $(p,l)$-capacity, inbedding theorems, and the spectrum of a~selfadjoint elliptic operator
Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 357-387.

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

Necessary and sufficient conditions are found for continuity, compactness, and closability of imbedding operators of some function spaces into the space $L_p$. These results (for $p=2$) give criteria for positive definiteness and discreteness of the spectrum of the Dirichlet problem for a selfadjoint elliptic operator of arbitrary order. Some integral inequalities are considered for differentiable functions on a cube.
@article{IM2_1973_7_2_a7,
     author = {V. G. Maz'ya},
     title = {On $(p,l)$-capacity, inbedding theorems, and the spectrum of a~selfadjoint elliptic operator},
     journal = {Izvestiya. Mathematics },
     pages = {357--387},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {1973},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1973_7_2_a7/}
}
TY  - JOUR
AU  - V. G. Maz'ya
TI  - On $(p,l)$-capacity, inbedding theorems, and the spectrum of a~selfadjoint elliptic operator
JO  - Izvestiya. Mathematics 
PY  - 1973
SP  - 357
EP  - 387
VL  - 7
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1973_7_2_a7/
LA  - en
ID  - IM2_1973_7_2_a7
ER  - 
%0 Journal Article
%A V. G. Maz'ya
%T On $(p,l)$-capacity, inbedding theorems, and the spectrum of a~selfadjoint elliptic operator
%J Izvestiya. Mathematics 
%D 1973
%P 357-387
%V 7
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1973_7_2_a7/
%G en
%F IM2_1973_7_2_a7
V. G. Maz'ya. On $(p,l)$-capacity, inbedding theorems, and the spectrum of a~selfadjoint elliptic operator. Izvestiya. Mathematics , Tome 7 (1973) no. 2, pp. 357-387. http://geodesic.mathdoc.fr/item/IM2_1973_7_2_a7/

[1] Mazya V. G., “O zadache Dirikhle dlya ellipticheskikh uravnenii proizvolnogo poryadka v neogranichennykh oblastyakh”, Dokl. AN SSSR, 450:6 (1963), 1221–1224

[2] Mazya V. G., “Klassy mnozhestv i mer, svyazannye s teoremami vlozheniya”, Teoremy vlozheniya i ikh prilozheniya (Baku, 1966), Tr. simpoziuma po teoremam vlozheniya, M., 1970, 142–469

[3] Molchanov A. M., “Ob usloviyakh diskretnosti spektra samosopryazhennykh uravnenii vtorogo poryadka”, Tr. Mosk. matem. ob-va, 2, 1953, 169–200

[4] Birman M. Sh., Pavlov B. S., “O polnoi nepreryvnosti nekotorykh operatorov vlozheniya”, Vestn. Leningr. un-ta, ser. matem., 1961, no. 1, 61–74 | MR | Zbl

[5] Mazya V. G., Khavin V. P., “Nelineinyi analog nyutonovskogo potentsiala i metricheskie svoistva $(p,l)$-emkosti”, Dokl. AN SSSR, 194:4 (1970), 770–773

[6] Sobolev S. L., “Ob odnoi teoreme funktsionalnogo analiza”, Matem. sb., 4(46) (1938), 471–497 | Zbl

[7] Ilin V. P., “Teorema vlozheniya v sluchae maksimalnogo pokazatelya”, Dokl. AN SSSR, 96 (1954), 905–908 | MR

[8] Mazya V. G., “Ob ustranimykh osobennostyakh ogranichennykh reshenii kvazilineinykh ellipticheskikh uravnenii lyubogo poryadka”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 6, Zapiski nauchnykh seminarov LOMI, 27, 1972, 116–130 | MR | Zbl

[9] Yafaev D. R., “O spektre vozmuschennogo poligarmonicheskogo operatora”, Problemy matem. fiziki, 5, 1971, 122–128 | Zbl

[10] Mazya V. G., Khavin V. P., “Nelineinaya teoriya potentsiala”, Uspekhi matem. nauk, 27:6 (1972), 67–138 | MR

[11] Aronszajn N., Smith K. T., “Theory of Bessel potentials, I”, Ann. Inst. Fourier, 11 (1961), 385–475 | MR | Zbl

[12] Fikhtengolts T. M., Kypc differentsialnogo i integralnogo ischisleniya, I, Fizmatgiz, M., 1958