On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems

Agmon, Shmuel

doi:10.1002/cpa.3160150203

Cited by 506 publications

(358 citation statements)

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“…Conditions 1), 2) were introduced by Agmon in [8], where the completeness of the system of eigen and associated (root) vectors was proved for an elliptic boundary problem in a smooth domain. These conditions appear in the study of an elliptic boundary problem with a parameter.…”

Section: Rays Of Minimal Growthmentioning

confidence: 99%

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On completeness of root functions of elliptic boundary problems in a domain with conical points on the boundary

Egorov

Kondratiev

Schulze

2002

Comptes Rendus. Mathématique

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In this paper 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 for a finite number of points such that each point has a neighborhood where the boundary is a conical surface.The problem of completeness of the system of eigen and associated functions of boundary value problems for elliptic operators in domains with smooth boundaries has been studied by numerous authors. F.B. Browder [1]-[3] obtained the theorem for the Dirichlet problem for elliptic operators of any order with a real principal part.Earlier M.V. Keldysh [4] proved the general theorem on the completeness of the system of eigen and associated functions of differential not self-adjoint operators and obtained as its corollary the theorem on the completeness for elliptic operators of second order with Dirichlet boundary conditions. For the Dirichlet problem for strongly elliptic differential operators of order 2m the completeness of the system of eigen and associated functions in L 2 (Ω), where Ω is an arbitrary bounded domain, was proved by M.S. Agranovich [5]. He studied also the problem with Neumann conditions, for the case of a Lipschitz boundary ∂Ω. The problem for elliptic systems of second order was studied by N.M. Krukovsky [6].All these authors referred to the methods of M.V. Keldysh [4]. We also use them here, together with the approach of T. Carleman as in [7].S. Agmon [8] and M. Schechter [9] proved that the system of root functions of an elliptic boundary problem is complete in a bounded domain Ω with a smooth

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Section: Rays Of Minimal Growthmentioning

confidence: 99%

“…Agmon [8] and M. Schechter [9] proved that the system of root functions of an elliptic boundary problem is complete in a bounded domain Ω with a smooth…”

mentioning

confidence: 99%

On completeness of root functions of elliptic boundary problems in a domain with conical points on the boundary

Egorov

Kondratiev

Schulze

2002

Comptes Rendus. Mathématique

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“…(note that |τ | ≈ C on γ (1) and |τ | ≈ CΛ on γ (2) ). Now we have to estimate |M j (ω ′ , τ, Λ)| in (4.4).…”

Section: M)mentioning

confidence: 99%

“…In the case µ = 0 this is the usual definition of ellipticity with parameter which was introduced by Agmon [1] and Agranovich-Vishik [3]. Therefore we may assume in the following that µ > 0.…”

Section: Introductionmentioning

confidence: 99%