Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Agoshkov and V. I. Lebedev, “Poincaré–Steklov operators and domain division methods in variational problems,” Vychisl. Proc. Sist., 2, 173–226 (1985).
M. S. Agranovich, “Spectral problems for second-order strongly elliptic systems in smooth and nonsmooth domains,” Russ. Math. Surv., 57, No. 5, 847–920 (2002).
M. S. Agranovich, “Remarks on potential spaces and Besov spaces in a Lipschitz domain and on Whitney arrays on its boundary,” Russ. J. Math. Phys., 15, No. 2, 146–155 (2008).
M. S. Agranovich, “Mixed problems in a Lipschitz domain for strongly elliptic second-order systems,” Funct. Anal. Appl., 45, No. 2, 81–98 (2011).
M. S. Agranovich, “Spectral problems in Lipschitz domains,” J. Math. Sci. (N. Y.), 190, 8–33 (2011).
M. S. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in Domains with Smooth and Lipschitz Boundaries [in Russian], MCNMO, Moscow (2013).
M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich, Generalized Method of Eigenoscillations in Difraction Theory, Wiley-VCH, Berlin (1999).
J.-P. Aubin, “Abstract boundary-value operators and their adjoint,” Rend. Semin. Mat. Univ. Padova, 43, 1–33 (1970).
J.-P. Aubin, Approximation of Elliptic Boundary-Value Problems, Courier Corporation, New York (2007).
Yu. M. Berezansky, Eigenfunction Decomposition of Self-Adjoint Operators [in Russian], Naukova dumka, Kiev (1965).
E. Gagliardo, “Caratterizazioni delle tracce sullo frontiera relative ad alcune classi de funzioni in “n” variabili,” Rend. Semin. Mat. Univ. Padova, 27, 284–305 (1957).
H. Gajewski, K. Greger, and K. Zacharias, Nichtlineare Operator Gleichungen und Operator Differential Gleichungen, Akademie-Verlag, Berlin (1974).
N. D. Kopachevsky, “On abstract Green formula for a triple of Hilbert spaces and its applications to the Stokes problem,” Tavr. Vestn. Inform. Mat., 2, 52–80 (2004).
N. D. Kopachevsky, “Abstract Green formula for mixed boundary value problems,” Uch. Zap. Tavr. Nat. Univ. im. V. I. Vernadskogo. Ser. Mat. Mekh. Inform. Kibern., 20, No. 2, 3–12 (2007).
N. D. Kopachevsky, “On abstract Green formula for mixed boundary value problems and some of its applications,” Spectr. Evol. Probl., 21, No. 1, 2–39 (2011).
N. D. Kopachevsky and S. G. Krein, “Abstract Green formula for a triple of Hilbert spaces, abstract boundary value and spectral problems,” Ukr. Math. Bull., 1, No. 1, 69–97 (2004).
N. D. Kopachevsky, S. G. Krein, and Ngo Zui Kan, Operator Methods in Linear Hydrodynamics: Evolution and Spectral Problems [in Russian], Nauka, Moscow (1989).
S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).
V. I. Lebedev and V. I. Agoshkov, Poncaré–Steklov Operators and Their Applications in Analysis [in Russian], Otd. Vychisl. Mat. AN SSSR, Moscow (1983).
J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogenes et Applications, Dunod, Paris (1968).
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Univ. Press, Cambridge (2000).
O. A. Oleinik, G. A. Iosifian, and A. S Shamaev, Mathematical Problems of Theory of Strongly Nonhomogeneous Elastic Media [in Russian], MGU, Moscow (1990).
B. V. Palcev, “Mixed problems with nonhomogeneous boundary conditions in Lipschitz domains for second-order elliptic equations with a parameter,” Sb. Math., 187, No. 4, 525–580 (1996).
V. S. Rychkov, “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains,” J. Lond. Math. Soc., 60, No. 1, 237–257 (1999).
W. McLean, Hilbert Space Methods for Partial Differential Equations, Electronic Monographs in Differential Equations, San Marcos (1994).
F. Sjarle, Finite Element Method for Elliptic Problems [Russian translation], Mir, Moscow (1980).
S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], Nauka, Moscow (1988).
L. R. Volevich and S. G. Gindikin, Generalized Functions and Convolution Equations [in Russian], Nauka, Moscow (1994).
V. I. Voytitsky, N. D. Kopachevsky, and P. A. Starkov, “Multicomponent problems of conjunction and the auxiliary abstract boundary problems,” J. Math. Sci. (N. Y.), 170, No. 2, 131–172 (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 57, Proceedings of the Crimean Autumn Mathematical School-Symposium KROMSH–2014, 2015.
Rights and permissions
About this article
Cite this article
Kopachevsky, N.D. Abstract Green Formulas for Triples of Hilbert Spaces and Sesquilinear Forms. J Math Sci 225, 226–264 (2017). https://doi.org/10.1007/s10958-017-3470-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-017-3470-9