On completeness of root functions of elliptic boundary problems in a domain with conical points on the boundary
Comptes Rendus. Mathématique, Volume 334 (2002) no. 8, pp. 649-654.

We prove the completeness of the system of eigen and associated functions (i.e., root functions) of an elliptic boundary value problem in a domain, whose boundary is a smooth surface everywhere, except at a finite number of points, such that each point possesses a neighborhood, where the boundary is a conical surface.

On montre que les fonctions propres et associées d'un problème au bord pour un opérateur elliptique d'ordre 2m, défini dans un domaine dans n avec points coniques sur le bord, forment un système total.

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DOI: 10.1016/S1631-073X(02)02320-8
Egorov, Youri V. 1; Kondratiev, Vladimir A. 2; Schulze, Bert-Wolfgang 3

1 Laboratoire des mathématiques pour l'industrie et la physique, UMR 5640, Université Paul Sabatier, UFR MIG, 118, route de Narbonne, 31062, Toulouse cedex 4, France
2 Mehmat Faculty, Lomonosov University, Vorob'evy Gory, 119899 Moscow, Russia
3 Institute of Mathematics, Potsdam University, 601553 Potsdam, Germany
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Egorov, Youri V.; Kondratiev, Vladimir A.; Schulze, Bert-Wolfgang. On completeness of root functions of elliptic boundary problems in a domain with conical points on the boundary. Comptes Rendus. Mathématique, Volume 334 (2002) no. 8, pp. 649-654. doi : 10.1016/S1631-073X(02)02320-8. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02320-8/

[1] Agmon, S. On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., Volume 15 (1962), pp. 119-147

[2] Agmon, S.; Douglis, A.; Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math., Volume 12 (1959), pp. 623-727

[3] Agranovich, M.S. Elliptic boundary problems, Partial Differential Equations IX, Encyclopedia of Mathematical Sciences, 79, Springer, 1991, pp. 1-144

[4] Agranovich, M.S. On series with respect to root vectors of operators associated with forms having symmetric principal part, Funct. Anal. Appl., Volume 28 (1994) no. 3, pp. 151-167

[5] Agranovich, M.S.; Denk, R.; Faerman, M. Weakly smooth nonselfadjoint spectral problems for elliptic boundary value problems (Demuth, P.; Schrohe, E.; Schulze, B.-W., eds.), Spectral Theory, Microlocal Analysis, Singular Manifolds, Birkhäuser, 2000, pp. 138-199

[6] Browder, F.E. On the eigenfunctions and eigenvalues of the general elliptic differential operator, Proc. Nat. Acad. Sci. USA, Volume 39 (1953), pp. 433-439

[7] Browder, F.E. Estimates and existence theorems for elliptic boundary value problems, Proc. Nat. Acad. Sci. USA, Volume 45 (1959), pp. 365-372

[8] Browder, F.E. On the spectral theory of strongly elliptic differential operators, Proc. Nat. Acad. Sci. USA, Volume 45 (1959), pp. 1423-1431

[9] Carleman, T. Über die Verteilung der Eigenwerte partieller Differentialgleichungen, Ber. der Sächs. Akad. Wiss. Leipzig, Mat. Nat. Kl., Volume 88 (1936), pp. 119-132

[10] Dunford, N.; Schwartz, J.T., Linear Operators, II, Interscience, New York, 1963

[11] Egorov, Yu.V.; Schulze, B.-W. Pseudo-Differential Operators, Singularities, Applications, Oper. Theory Adv. Appl., 93, Birkhäuser, 1997

[12] Keldysh, M.V. On the eigenvalues and eigenfunctions of certain classes of non-selfadjoint equations, Dokl. AN SSSR, Volume 77 (1951), pp. 11-14

[13] Kondratiev, V.A. Boundary value problems for elliptic equations in domains with conical or singular points, Trudy Moskov. Mat. Obshch., Volume 16 (1967), pp. 209-292

[14] Krukovsky, N.M. Theorems on the m-fold completeness of the generalized eigen- and associated functions from W21 of certain boundary value problems for elliptic equations and systems, Differentsial'nye Uravneniya, Volume 12 (1976) no. 10, pp. 1842-1851

[15] Schechter, M. Remarks on elliptic boundary value problems, Comm. Pure Appl. Math., Volume 12 (1959), pp. 457-482

[16] Schulze, B.-W. Pseudo-Differential Operators on Manifolds with Singularities, North-Holland, Amsterdam, 1991

[17] Schulze, B.-W. Boundary Value Problems and Singular Pseudo-Differential Operators, Wiley, Chichester, 1998

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