Abstract
In this paper we give, for the first time, an abstract interpretation of nonlocal boundary value problems for elliptic differential equations of the second order. We prove coerciveness and Fredholmness of nonlocal boundary value problems for the second order elliptic differential-operator equations. We apply then, in section 6, these results for investigation of nonlocal boundary value problems for the second order elliptic differential equations (one can find the references on the subject in the introduction and Chapter V in the book by A. L. Skubachevskii [27]). Abstract results obtained in this paper can be used for study of nonlocal boundary value problems for quasielliptic differential equations.
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References
Agmon, S.,On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math.,15 (1962), 119–147.
Agmon, S. and Nirenberg, L.,Properties of solutions of ordinary differential equations in Banach spaces, Comm. Pure Appl. Math.,16 (1963), 121–239.
Agranovich, M. S. and Vishik, M. I.,Elliptic problems with a parameter and parabolic problems of general type, Uspekhi Mat. Nauk,19, 3 (1964), 53–161 (Russian) English translation in Russian Math. Surveys,19, 3 (1964), 53–159).
Aibeche, A.,Coerciveness estimates for a class of nonlocal elliptic problems, Differential Equations and Dynamical Systems,1, 4 (1993), 341–351.
Aliev, I. V.,Differential-operator equations with irregular boundary conditions and their applications, Differen. Uravnen.,30, 1 (1994), 95–103.
Amann, H.,Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math.,45 (1983), 225–254.
deLaubenfels, R.,Incomplete iterated Cauchy problems, J. Math. Anal. and Appl.,168, 2 (1992), 552–579.
Dore G. and Yakubov, S.,Semigroup estimates and noncoercive boundary value problems, Semigroup Forum, 1–30 (to appear).
Dunford, N. and Schwartz, J. T., “Linear Operators. Part II. Spectral Theory”, Interscience, New York, 1963.
Favini, A.,Su un problema ai limiti per certe equazioni astratte del secondo ordine, Rend. Sem. Mat. Univ. Padova,53 (1975), 211–230.
Gorbachuk, V. I. and Gorbachuk, M. L., “Boundary Value Problems for Differentian-tial-Operator Equations”, Naukova Dumka, Kiev, 1984.
Grisvard, P.,Caracterization de quelques espaces d'interpolation, Arch. Rat. Mach. Anal.,25 (1967), 40–63.
Grisvard, P.,Equations differentielles abstraites, Ann. Sci. Ecolo Norm. Sup., 4e, 2 (1969), 311–395.
Grisvard, P., “Elliptic Problems in Nonsmooth Domains”, Pitman, Boston, 1985.
Karakas, H. I., Shakhmurov V. B. and Yakubov, S.,Degenerate elliptic boundary value problems, Applicable Analysis,60 (1996), 155–174.
Kondratiev, V. A.,Boundary value problems for elliptic equations in domains with conic or corner points, Trans. Moscow. Math. Soc., (1967), 227–313.
Kondratiev, V. A. and Oleinik O. A.,Boundary value problems for partial differential equations in non-smooth domains, Russian. Math. Surveys,38, 2 (1983), 1–86.
Krein, S. G., “Linear Differential Equations in Banach Space”, Providence, 1971.
Krein, S. G., “Linear Equations in Banach Space”, Birkhauser, 1982.
Labbas, R. and Terreni, B.,Sommes d'operateurs de type elliptique et paraboli-que. 2e partie: Applications, Bollettino U. M. I.,2-B, 7 (1988), 141–162.
Lions, J. L. and Peetre, J.,Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Sci. Publ. Math.,19 (1964), 5–8.
Nazarov, S. A. and Plamenevskii, B. A., “Elliptic Problems in Domains with Piecewise Smooth Boundaries”, Walter de Gruyter, Berlin, New York, 1994.
Schwartz, J. T.,A remark on inequalities of Calderon-Zygmund type for vector-valued functions, Comm. Pure Appl. Math.,14 (1961), 785–799.
Seeley, R.,Fractional powers of boundary problems, Actes, Congres. Intern. Math. 1970,2 (1971), 795–801.
Seeley, R.,Interpolation in L p with boundary conditions, Studia Math.,44 (1972), 47–60.
Shklyar, A. Ya., “Complete Second Order Linear Differential Equations in Hilbert Spaces”, Birkhäuser Verlag, Basel, 1997.
Skubachevskii, A. L., “Elliptic Functional Differential Equations and Applications”, Birkhäuser Verlag, Basel, 1997.
Triebel, H., “Interpolation Theory. Function Spaces. Differential Operators”, North-Holland, Amsterdam, 1978.
Yakubov, S., “Completeness of Root Functions of Regular Differential Operators”, Longman, Scientific and Technical, New York, 1994.
Yakubov S. and Yakubov Ya., “Differential-Operator Equations. Ordinary and Partial Differential Equations”, (to appear).
Yakubov, S. Ya. and Aliev, B. A.,Boundary value problem with an operator in the boundary conditions for an elliptic differential-operator equation of the second order, Sibir. Math. J.,26, 4 (1985), 176–188.
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Yakubov, S. A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integr equ oper theory 35, 485–506 (1999). https://doi.org/10.1007/BF01228044
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DOI: https://doi.org/10.1007/BF01228044