Boundary properties of solutions of differential equations and general boundary-value problems

Burskiĭ, V.

doi:10.1090/S0077-1554-07-00162-8

Boundary properties of solutions of differential equations and general boundary-value problems
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by V. P. Burskiĭ
Translated by: G. G. Gould

Trans. Moscow Math. Soc. 2007, 163-200

DOI: https://doi.org/10.1090/S0077-1554-07-00162-8

Published electronically: October 29, 2007

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Abstract:

For a general differential operator with smooth matrix-valued coefficients in a bounded domain with smooth boundary we consider the boundary properties of functions from the domain of definition of a maximal extension in $L_2(\Omega )$ and we study the properties of extensions and boundary-value problems corresponding to them. The investigations are based on Green’s formula.

References

M. S. Agranovič, On partial differential equations with constant coefficients, Uspehi Mat. Nauk 16 (1961), no. 2 (98), 27–93 (Russian). MR 0133597
Ju. M. Berezans′kiĭ, Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Izdat. “Naukova Dumka”, Kiev, 1965 (Russian). Akademijá Nauk Ukrainskoĭ SSSR. Institut Matematiki. MR 0222719
N. Bourbaki, Topological vector spaces. Chapters 1–5, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1987. Translated from the French by H. G. Eggleston and S. Madan. MR 910295, DOI 10.1007/978-3-642-61715-7
V. P. Burskiĭ, Harmonic analysis in boundary value problems for partial differential equations with constant coefficients, Dokl. Akad. Nauk Ukrain. SSR Ser. A 3 (1986), 7–10, 90 (Russian, with English summary). MR 842785
V. P. Burskiĭ, Boundary properties of $L_2$-solutions of linear differential equations and the equation-domain duality, Dokl. Akad. Nauk SSSR 309 (1989), no. 5, 1036–1039 (Russian); English transl., Soviet Math. Dokl. 40 (1990), no. 3, 592–595. MR 1037114
V. P. Burskiĭ, Uniqueness of the solution of the Dirichlet problem in a ball for the wave equation, Differentsial′nye Uravneniya 24 (1988), no. 6, 1038–1039, 1101 (Russian). MR 953854
V. P. Burskiĭ, Remarks on the Dirichlet problem for an ultrahyperbolic equation and on integral geometry on the sphere, Uspekhi Mat. Nauk 43 (1988), no. 5(263), 181–182 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 5, 215–216. MR 971474, DOI 10.1070/RM1988v043n05ABEH001938
V. P. Burskiĭ, Boundary value problems for a second-order hyperbolic equation in a disk, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (1987), 22–29, 83 (Russian). MR 889192
V.P. Burskiĭ, Methods of investigating boundary-value problems for general differential equations, Naukova Dumka, Kiev, 2002. (Russian)
V. P. Burskiĭ, Generalized solutions of boundary value problems for differential equations of general type, Uspekhi Mat. Nauk 53 (1998), no. 4(322), 215–216 (Russian); English transl., Russian Math. Surveys 53 (1998), no. 4, 864–865. MR 1668066, DOI 10.1070/rm1998v053n04ABEH000061
V. P. Burskiĭ, A commutative diagram connected with a differential operator in a domain, Ukrain. Mat. Zh. 43 (1991), no. 12, 1703–1709 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 43 (1991), no. 12, 1588–1594 (1992). MR 1172313, DOI 10.1007/BF01066700
V. P. Burskiĭ, On the uniqueness of the solution of some boundary value problems for differential equations in a domain with an algebraic boundary, Ukraïn. Mat. Zh. 45 (1993), no. 7, 898–906 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 45 (1993), no. 7, 993–1003 (1994). MR 1260648, DOI 10.1007/BF01057446
V. P. Burskiĭ, Boundary value problems for an elliptic equation with complex coefficients and a moment problem, Ukraïn. Mat. Zh. 45 (1993), no. 11, 1476–1483 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 45 (1993), no. 11, 1659–1668 (1994). MR 1307364, DOI 10.1007/BF01060856
V. P. Burskiĭ, Solutions of the Dirichlet problem for elliptic systems in a disk, Ukraïn. Mat. Zh. 44 (1992), no. 10, 1307–1313 (Russian, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 44 (1992), no. 10, 1197–1203 (1993). MR 1201128, DOI 10.1007/BF01057674
V.P. Burskiĭ, On the kernel of a differential operator with constant lower-order coefficients in a disk, Manuscript deposited in VINITI, no. 3796–82 Dep. (Russian)
V. P. Burskiĭ, Theorems on traces of the solution of the string vibration equation in the disk, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 23 (1985), 35 (Russian). MR 830507
L. Ĭ. Vaĭnerman, Extensions of closed operators in Hilbert space, Mat. Zametki 28 (1980), no. 6, 833–842, 960 (Russian). MR 603218
M. I. Višik, On general boundary problems for elliptic differential equations, Trudy Moskov. Mat. Obšč. 1 (1952), 187–246 (Russian). MR 0051404
V. I. Gorbachuk and M. L. Gorbachuk, Granichnye zadachi dlya differentsial′no-operatornykh uravneniĭ, “Naukova Dumka”, Kiev, 1984 (Russian). MR 776604
V. I. Gorbachuk, M. L. Gorbachuk, and A. N. Kochubeĭ, The theory of extensions of symmetric operators, and boundary value problems for differential equations, Ukrain. Mat. Zh. 41 (1989), no. 10, 1299–1313, 1436 (Russian); English transl., Ukrainian Math. J. 41 (1989), no. 10, 1117–1129 (1990). MR 1034669, DOI 10.1007/BF01057246
M. L. Gorbačuk, Selfadjoint boundary value problems for a second order differential equation with an unbounded operator coefficient, Funkcional. Anal. i Priložen. 5 (1971), no. 1, 10–21 (Russian). MR 0283624
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1009163
A. A. Dezin, Obshchie voprosy teorii granichnykh zadach, “Nauka”, Moscow, 1980 (Russian). MR 596223
Yu. V. Egorov and M. A. Shubin, Linear partial differential equations. Foundations of the classical theory, Partial differential equations, 1 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 5–265 (Russian). MR 1141629
M. Krein, The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I, Rec. Math. [Mat. Sbornik] N.S. 20(62) (1947), 431–495 (Russian, with English summary). MR 0024574
S. G. Kreĭn, Lineĭ nye uravneniya v banakhovom prostranstve, Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0374949
A. N. Kočubeĭ, Extensions of symmetric operators and of symmetric binary relations, Mat. Zametki 17 (1975), 41–48 (Russian). MR 365218
L. P. Kupcov, The mean value property and the maximum principle for second order parabolic equations, Dokl. Akad. Nauk SSSR 242 (1978), no. 3, 529–532 (Russian). MR 507137
O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
V. È. Lyantse and O. G. Storozh, Metody teorii neogranichennykh operatorov, “Naukova Dumka”, Kiev, 1983 (Russian). MR 757535
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
A. H. Mamjan, The construction of solvable extensions in a parallelepiped for linear differential operators with constant coefficients, Differencial′nye Uravnenija 6 (1970), 358–370 (Russian). MR 0279426
M. A. Naĭmark, Linear differential operators. Part II: Linear differential operators in Hilbert space, Frederick Ungar Publishing Co., New York, 1968. With additional material by the author, and a supplement by V. È. Ljance; Translated from the Russian by E. R. Dawson; English translation edited by W. N. Everitt. MR 0262880
V. P. Palamodov, Linear differential operators with constant coefficients, Die Grundlehren der mathematischen Wissenschaften, Band 168, Springer-Verlag, New York-Berlin, 1970. Translated from the Russian by A. A. Brown. MR 0264197
B. P. Panejah, General systems of differential equations with constant coefficients. , Dokl. Akad. Nauk SSSR 138 (1961), 297–300 (Russian). MR 0126606
B. I. Ptashnik, Nekorrektnye granichnye zadachi dlya differentsial′nykh uravneniĭ s chastnymi proizvodnymi, “Naukova Dumka”, Kiev, 1984 (Russian). MR 772024
Ja. A. Roĭtberg, The existence of limit values of generalized solutions of elliptic equations on the boundary of the domain, Sibirsk. Mat. Zh. 20 (1979), no. 2, 386–396, 460 (Russian). MR 530503
Ja. A. Roĭtberg, On the boundary values of generalized solutions of systems that are elliptic in the sense of Douglis and Nirenberg, Sibirsk. Mat. Ž. 18 (1977), no. 4, 846–860, 957 (Russian). MR 0486967
Ya. A. Roĭtberg and V. A. Serdyuk, Elliptic problems with a parameter in $L_{2}$-spaces of generalized functions for general systems of equations, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 30 (1982), 62 (Russian). MR 692462
F. S. Rofe-Beketov, Selfadjoint extensions of differential operators in a space of vector-valued functions, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 8 (1969), 3–24 (Russian). MR 0281055
I. V. Skrypnik and I. V. Skrypnik, Nelineĭnye èllipticheskie uravneniya vysshego poryadka, Izdat. “Naukova Dumka”, Kiev, 1973 (Russian). MR 0435590
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
Lars Hörmander, Differential operators of principal type, Math. Ann. 140 (1960), 124–146. MR 130574, DOI 10.1007/BF01360085
Lars Hörmander, The analysis of linear partial differential operators. I, 2nd ed., Springer Study Edition, Springer-Verlag, Berlin, 1990. Distribution theory and Fourier analysis. MR 1065136, DOI 10.1007/978-3-642-61497-2
Lars Hörmander, The analysis of linear partial differential operators. II, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR 705278, DOI 10.1007/978-3-642-96750-4
V. P. Burskii, On well-posedness of boundary value problems for some class of general PDEs in a generalized setting, Funct. Differ. Equ. 8 (2001), no. 1-2, 89–100. International Conference on Differential and Functional Differential Equations (Moscow, 1999). MR 1949991
J. W. Calkin, Abstract symmetric boundary conditions, Trans. Amer. Math. Soc. 45 (1939), no. 3, 369–442. MR 1501997, DOI 10.1090/S0002-9947-1939-1501997-7
Avron Douglis and Louis Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8 (1955), 503–538. MR 75417, DOI 10.1002/cpa.3160080406
Herbert Gajewski, Konrad Gröger, and Klaus Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38, Akademie-Verlag, Berlin, 1974 (German). MR 0636412
Gerd Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 425–513. MR 239269
Gudrun Gudmundsdottir, Global properties of differential operators of constant strength, Ark. Mat. 15 (1977), no. 2, 169–198. MR 509109, DOI 10.1007/BF02386038
V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1–95. MR 1087947, DOI 10.1016/0022-1236(91)90024-Y
M. M. Malamud and V. I. Mogilevskii, Kreĭn type formula for canonical resolvents of dual pairs of linear relations, Methods Funct. Anal. Topology 8 (2002), no. 4, 72–100. MR 1942823
Lars Hörmander, Definitions of maximal differential operators, Ark. Mat. 3 (1958), 501–504. MR 106333, DOI 10.1007/BF02589511
Bent Fuglede, A priori inequalities connected with systems of partial differential equations, Acta Math. 105 (1961), 177–195. MR 140818, DOI 10.1007/BF02559589