Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms

N. D. Kopachevsky

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 57 (2015), pp. 71-107.

Résumé

In this paper, under several general assumptions, we deduce an abstract Green formula for a triple of Hilbert spaces and an (abstract) trace operator and a similar formula corresponding to sesquilinear forms. We establish existence conditions for the abstract Green formula for mixed boundary-value problems. As the main application, we deduce generalized Green formulas for the Laplace operator applied to boundary-value problems in Lipschitz domains.

Bibliographie

[1] Agoshkov V. I., Lebedev V. I., “Operatory Puankare—Steklova i metody razdeleniya oblasti v variatsionnykh zadachakh”, Vychisl. prots. i sist., 2 (1985), 173–226 | MR

[2] Agranovich M. S., “Spektralnye zadachi dlya silno ellipticheskikh sistem vtorogo poryadka v oblastyakh s gladkoi i negladkoi granitsei”, Usp. mat. nauk, 57:5 (2002), 3–78 | DOI | MR | Zbl

[3] Agranovich M. S., “Smeshannye zadachi v lipshitsevoi oblasti dlya silno ellipticheskikh sistem 2-go poryadka”, Funkts. analiz i ego prilozh., 45:2 (2011), 1–22 | DOI | MR | Zbl

[4] Agranovich M. S., “Spektralnye zadachi v lipshitsevykh oblastyakh”, Sovrem. mat. Fundam. napravl., 39, 2011, 11–35 | MR

[5] Agranovich M. S., Sobolevskie prostranstva, ikh obobscheniya i ellipticheskie zadachi v oblastyakh s gladkoi i lipshitsevoi granitsei, MTsNMO, M., 2013

[6] Berezanskii Yu. M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova dumka, Kiev, 1965 | MR

[7] Veil G., “Metod ortogonalnoi proektsii v teorii potentsiala”, Izbrannye trudy. Matematika i teoreticheskaya fizika, Nauka, M., 1984, 275–307

[8] Voytitsky V. I., Kopachevsky N. D., Starkov P. A., “Multicomponent conjugation (transmission) problems and auxillary abstract boundary-value problems”, J. Math. Sci., 170:2 (2010), 131–172 | DOI | MR

[9] Volevich L. R., Gindikin S. G., Obobschennye funktsii i uravneniya v svertkakh, Nauka, M., 1994 | MR

[10] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR

[11] Kopachevskii N. D., “Ob abstraktnoi formule Grina dlya troiki gilbertovykh prostranstv i ee prilozheniyakh k zadache Stoksa”, Tavricheskii vestn. inform. i mat., 2 (2004), 52–80

[12] Kopachevskii N. D., “Abstraktnaya formula Grina dlya smeshannykh kraevykh zadach”, Uchenye zapiski Tavricheskogo natsionalnogo un-ta im. V. I. Vernadskogo. Seriya “Matematika. Mekhanika. Informatika i kibernetika”, 20:2 (2007), 3–12

[13] Kopachevskii N. D., “Ob abstraktnoi formule Grina dlya smeshannykh kraevykh zadach i nekotorykh ee prilozheniyakh”, Spektr. i evolyuts. zadachi, 21:1 (2011), 2–39

[14] Kopachevskii N. D., Krein S. G., Ngo Zui Kan, Operatornye metody v lineinoi gidrodinamike: evolyutsionnye i spektralnye zadachi, Nauka, M., 1989 | MR

[15] Kopachevskii N. D., Krein S. G., “Abstraktnaya formula Grina dlya troiki gilbertovykh prostranstv, abstraktnye kraevye i spektralnye zadachi”, Ukr. mat. vestn., 1:1 (2004), 69–97 | MR | Zbl

[16] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[17] Lebedev V. I., Agoshkov V. I., Operatory Puankare–Steklova i ikh prilozheniya v analize, Otd. vychisl. matem. AN SSSR, M., 1983 | MR

[18] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971

[19] Paltsev B. V., “O smeshannoi zadache s neodnorodnymi granichnymi usloviyami dlya ellipticheskikh s parametrom uravnenii vtorogo poryadka v lipshitsevykh oblastyakh”, Mat. sb., 187:4 (1996), 59–116 | DOI | MR | Zbl

[20] Oben Zh.-P., Priblizhennoe reshenie ellipticheskikh kraevykh zadach, Mir, M., 1977 | MR

[21] Oleinik O. A., Iosifyan G. A., Shamaev A. S., Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, MGU, M., 1990

[22] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988 | MR

[23] S'yarle F., Metod konechnykh elementov dlya ellipticheskikh zadach, Mir, M., 1980 | MR

[24] Agranovich M. S., “Remarks on potential spaces and Besov spaces in a Lipschitz domain and on Whitney arrays on its boundary”, Russ. J. Math. Phys., 15:2 (2008), 146–155 | DOI | MR | Zbl

[25] Agranovich M. S., Katsenelenbaum B. Z., Sivov A. N., Voitovich N. N., Generalized method of eigenoscillations in difraction theory, Wiley-VCH, Berlin–Toronto, 1999 | MR

[26] Aubin J.-P., “Abstract boundary-value operators and their adjoint”, Rend. Semin. Mat. Univ. Padova, 43 (1970), 1–33 | MR | Zbl

[27] Gagliardo E., “Caratterizazioni delle trace sullo frontiera relative ad alcune classi de funzioni in $n$ variabili”, Rend. Semin. Mat. Univ. Padova, 27 (1957), 284–305 | MR | Zbl

[28] McLean W., Strongly elliptic systems and boundary integral equations, Cambridge University Press, 2000 | MR | Zbl

[29] Rychkov V. S., “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains”, J. Lond. Math. Soc., 60:1 (1999), 237–257 | DOI | MR | Zbl

[30] Showalter R. E., Hilbert space methods for partial differential equations, Electronic Monographs in Differential Equations, San Marcos, TX, 1994 | MR