Complete Second Order Linear Differential Equations in Hilbert Spaces - Alexander Ya. Shklyar

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Résumé :
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Sommaire:
I. Well-posedness of boundary-value problems.- to Part I.- 1. Joint spectrum of commuting normal operators and its position. Estimates for roots of second order polynomials. Definition of well-posedness of boundary-value problems.- 1. Introductory notes.- 2. Joint spectrum of operators.- 3. Position of the joint spectrum.- 4. Estimates for roots of characteristic polynomials.- 5. Definitions of well-posedness and weak well-posedness of boundary-value problems for equation (1).- 6. Spaces of boundary data.- 7. (Weak) well-posedness and uniform (weak) well-posedness.- 2. Well-posedness of boundary-value problems for equation (1) in the case of commuting self-adjoint A and B.- 1. The Cauchy problem.- 2. The Dirichlet problem.- 3. The Neumann problem.- 4. The inverse Cauchy problem.- 3. The Cauchy problem.- 1. Distinction of the general case of commuting normal operators A and B.- 2. A criterion for the weak well-posedness.- 3. Proof of Theorem 3.1.- 4. A criterion for the well-posedness.- 5. The (weak) well-posedness in particular cases.- 6. Spectrum of the associated operator pencil.- 7. Fattorini's definitions of well-posedness of the Cauchy problem.- 4. Boundary-value problems on a finite segment.- 1. The (weak) well-posedness of the Dirichlet problem.- 2. The (weak) well-posedness of the Dirichlet problem in particular cases.- 3. Boundary conditions for the Dirichlet problem.- 4. The weak well-posedness and the well-posedness of the Neumann problem.- 5. Boundary conditions for the Neumann problem.- 6. The inverse Cauchy problem.- II. Initial data of solutions.- to Part II.- 5. Boundary behaviour of an integral transform R(t) as t ? 0 depending on the sub-integral measure.- 1. Analogy with Tauberian theorems.- 2. A model example.- 3. Proof of Lemma 5.1.- 4. Further results.- 5. Continuity of R(t) on R+. in extreme cases.- 6. Continuity of R(t) on R+.- 7. Continuity, boundedness, and integrability of R(t) on a finite segment [0,T].- 8. Equivalence of conditions on a sub-integral measure.- 6. Initial data of solutions.- 1. The set of initial data of solutions.- 2. When FC = D(B) x (D(A) ? D(