×

zbMATH — the first resource for mathematics

Duality in the spaces of solutions of elliptic systems. (English) Zbl 0919.35040
The aim of this paper is to investigate the structure of the strong dual of the space of solutions of a linear elliptic system \(Pu= 0\) on an open subset of \(\mathbb{R}^N\) (both determined and overdetermined elliptic systems are considered).
The main result proved in the paper is the following: Let \(X\) be an open set in \(\mathbb{R}^N\) and let the coefficients in \(P\) be real analytic on \(X\). Assume that \(D\subset\subset X\) is a domain with real analytic boundary and that \(U^*\subset U\) are neighbourhoods of the closure of \(D\). Then the space of solutions of \(Pu= 0\) on the closure of \(D\) is topologically equivalent to the dual of the space of solutions of \(Pu= 0\) on \(D\).

MSC:
35J45 Systems of elliptic equations, general (MSC2000)
35A20 Analyticity in context of PDEs
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] L.A. Aizenberg - S.G. Gindikin , On the general form of a linear continuous functional in spaces of holomorphic functions , Moscov. Oblast. Ped. Inst. Ucen. Zap 137 ( 1964 ), 7 - 15 (Russian). MR 180699
[2] P. Blanchet , Théorèmes de Fusion et de Dualité pour les Solutions d’Équations Elliptiques, Thése , Université de Montréal , Montréal , 1988 .
[3] A. Grothendieck , Sue les espaces de solutions d’une classe generale d’equations aux derivees partielles , J. Anal. Math. 2 ( 1952 -1953), 243 - 280 . MR 65811 | Zbl 0051.08801 · Zbl 0051.08801 · doi:10.1007/BF02825639
[4] V.P. Havin , Spaces of Analytic Functions, in: Mathematical analysis , VINITI , Moscow , 1966 , 76 - 164 . MR 206694
[5] L. Hörmander , An Introduction to Complex Analysis in Several Complex Variables , Van Nostrand , Princeton, NJ , 1966 . MR 203075 | Zbl 0138.06203 · Zbl 0138.06203
[6] G. Köthe , Topologische Lineare Räume, I. Srpinger-Verlag , Berlin et al., 1960 . MR 130551 | Zbl 0093.11901 · Zbl 0093.11901
[7] F. Mantovani - S. Spagnolo , Funzionali analitici e funzioni armoniche , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 ( 1964 ), 475 - 512 . Numdam | MR 174965 | Zbl 0134.11102 · Zbl 0134.11102 · numdam:ASNSP_1964_3_18_4_475_0 · eudml:83334
[8] M. Morimoto , An Introduction to Sato’s Hyperfunctions , AMS , Providence, Rhode Island , 1993 . Zbl 0811.46034 · Zbl 0811.46034
[9] C.B. Morrey - L. Nirenberg , On the analyticity of the solutions oflinear ellipctic systems of partial differential equations , Comm. Pure Appl. Math. 10 ( 1957 ), 271 - 290 . MR 89334 | Zbl 0082.09402 · Zbl 0082.09402 · doi:10.1002/cpa.3160100204
[10] M. Nacinovich - A.A. Shlapunov , On iterations of the green integrals an their applications to elliptic differential complexes , Math. Nachr. 180 ( 1996 ), 243 - 284 . MR 1397675 | Zbl 0871.35066 · Zbl 0871.35066 · doi:10.1002/mana.3211800111
[11] Ya A. Roitberg , Elliptic Boundary value Problems in Generalized Functions , Kluwer Academic Publishers , Dordrecht NL , 1995 (To appear). · Zbl 0937.35047
[12] A.A. Shlapunov - N.N. Tarkhanov , Bases with double orthogonality in the Cauchy problem for systems with injective symbols , Proc. London Math. Soc. 71 ( 1995 ), 1 - 52 . MR 1327932 | Zbl 0828.35040 · Zbl 0828.35040 · doi:10.1112/plms/s3-71.1.1
[13] E.L. Stout , Harmonic Duality, Hyperfunctions and Removable Singularities, Preprint , Univ. of Washington , Seattle , 1995 , p. 41 . MR 1481618 · Zbl 0876.32003
[14] N.N. Tarkhanov , Laurent Series for Solutions of Elliptic Systems , Nauka , Novosibirsk , 1991 , p. 317 (Russian). MR 1226897 | Zbl 0743.35021 · Zbl 0743.35021
[15] N.N. Tarkhanov , The Cauchy Problem for Solutions of Elliptic Equations , Akademie-Verlag , Berlin , 1995 . MR 1334094 | Zbl 0831.35001 · Zbl 0831.35001
[16] N.N. Tarkhanov , Complexes of Differential Operators , Kluwer Academic Publishers , Dordrecht, NL , 1995 . MR 1368856 | Zbl 0852.58076 · Zbl 0852.58076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.