Complete Second Order Linear Differential Equations in Hilbert Spaces

Abstract

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(|B|1/2))?.- 3. When FC = D(B) x D(A)?.- 4. E-sequences of vectors and the general expression for weak solutions.- 5. The set of initial data of weak solutions.- 6. When $${F_C}^\prime = H \times {H_{ - 1}}$$?.- 7. Fatou-Riesz property.- III. Extension, stability, and stabilization of weak solutions.- to Part III.- 7. The general form of weak solutions.- 1. Another general expression for weak solutions.- 2. Continuity, boundedness, and integrability of R(t) on [0.T] in a more general case.- 3. The general form of weak solutions where (2.2) holds.- 4. Initial data of weak solutions where (2.2) holds.- 5. Weak well-posedness of the Cauchy problem in a special space of initial data.- 8. Fatou-Riesz property.- 1. Fatou-Riesz property.- 2. Two-sided Fatou-Riesz property.- 3. First order equation and incomplete second order equations.- 4. The case of self-adjoint A and B.- 5. Spectrum of the associated operator pencil.- 9. Extension of weak solutions.- 1. Extension of weak solutions on a finite interval.- 2. Boundedness of weak solutions on a finite interval.- 3. Exponential growth of weak solutions.- 4. Two-sided extension of weak solutions.- 5. Spectrum of the associated operator pencil.- 6. Comparison of the results on extension of weak solutions and bounded weak solutions.- 7. Intermediate classes of weak solutions.- 8. Extension of weak solutions and weak well-posedness of boundary-value problems.- 10. Stability and stabilization of weak solutions.- 1. Stability and uniform stability of an equation.- 2. Stabilization of the Cesaro means for weak solutions.- 3. Stabilization of a weak solution.- 4. Stabilization of weak solutions and asymptotic stability of an equation.- 5. Exponential stability and uniform exponential stability of an equation.- 6. Stabilization of $$\frac{{y(t)}}{t}$$ for weak solutions of an equation.- 7. The case of self-adjoint A and B.- IV. Boundary-value problems on a half-line.- to Part IV.- 11. The Dirichlet problem on a half-line.- 1. Classes of (weak) uniqueness.- 2. Existence of (weak) solutions.- 3. A criterion for the (weak) well-posedness.- 4. Boundary data of solutions.- 12. The Neumann problem on a half-line.- 1. Classes of uniqueness (weak uniqueness).- 2. Existence of solutions and weak solutions.- 3. Criteria for the well-posedness and the weak well-posedness.- 4. Boundary data of solutions and weak solutions.- Commentaries on the literature.- List of symbols.