THE SOLVABILITY OF MIXED PROBLEMS FOR HYPERBOLIC AND PARABOLIC EQUATIONS

Abstract

CONTENTS Introduction Chapter 1. A survey of work on the mixed problem § 1. The formal scheme of the Fourier method § 2. A survey of results contained in the text-books § 3. Investigations relating to the wave equation § 4. The generalised solution of the general hyperbolic equation § 5. Further investigations of the hyperbolic equation § 6. The definition and uniqueness of the classical solution § 7. Solvability of the mixed problem for the hyperbolic equation in an arbitrary normal cylinder § 8. The justification of the Fourier method for the parabolic equation in a normal cylinder Chapter 2. Uniqueness of the classical solution in an arbitrary normal cylinder § 9. Uniqueness theorem for the weakly classical solution § 10. Existence of a finite energy for almost all t Chapter 3. Convergence of the basic bilinear series § 11. Summary of some results from the theory of elliptic equations § 12. Convergence of the basic bilinear series of eigenfunctions § 13. Convergence of the bilinear series of first derivatives § 14. Convergence of the bilinear series of second derivatives Chapter 4. Auxiliary results on the order of magnitude of the Fourier coefficients § 15. Two preliminary lemmas § 16. Basic lemmas on the order of magnitude of the Fourier coefficients Chapter 5. Solvability of the mixed problem for the hyperbolic equation in an arbitrary normal cylinder § 17. Proof of Theorem 8 § 18. Analysis of the conditions of Theorem 8 Chapter 6. The justification of the Fourier method for the parabolic equation in an arbitrary normal cylinder § 19. Proof of Theorem 9 References