The Boundary Value Problems of Mathematical Physics

Abstract

Preface to the English Edition I completed myself, and the translation of the" Introduction" and of this preface was done under the supervision of Springer-Verlag.I thank all who have worked on this edition, especially the editorial and production staff of Springer-Verlag. July 1984 o. A. LADYZHENSKAYA Contents Contents CHAPTER III Equations of Parabolic Type §1. Posing Initial-Boundary Value Problems and the Cauchy Problem §2. First Initial-Boundary Value Problem for the Heat Equation §3. First Initial-Boundary Value Problem for General Parabolic Equations §4. Other Boundary Value Problems. The Method of Fourier and Laplace. The Second Fundamental Inequality §5. The Method of Rothe Supplements and Problems CHAPTER IV Equations of Hyperbolic Type §l. General Considerations. Posing the Fundamental Problems §2. The Energy Inequality. Finiteness of the Speed of Propagation of Perturbations. Uniqueness Theorem for Solutions in w~ §3. The First Initial-Boundary Value Problem. Solvability in Wi(QT) §4. On the Smoothness of Generalized Solutions §5. Other Initial-Boundary Value Problems §6. The Functional Method of Solving Initial-Boundary Value Problems §7. The Methods of Fourier and Laplace Supplements and Problems CHAPTER V Some Generalizations §l. Elliptic Equations of Arbitrary Order. Strongly Elliptic Systems §2. Strongly Parabolic and Strongly Hyperbolic Systems §3. Schrodinger-Type Equations and Related Equations §4. Diffraction Problems Supplements and Problems CHAPTER VIThe Method of Finite Differences §1. General Description of the Method.

The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary

“…that corresponds to an eigenvalue λ ε P (ε) ≤ λ P + c p ε γ 0 and is orthogonal in L 2 (Ω ε ) to each of the eigenfunctions u ε j (1) , . .…”

Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

“…It is well-known (see [19]) that the initial boundary value problem (1.2) -(1.4) has a unique solution if the functions P, ρ, and f are smooth enough.…”

“…Formal differentiation of the series (3.17) gives, with the help of (3.18), 19) where the series converges uniformly. We further differentiate Eq-n (3.17) termwise twice; the new series converges in L 2 (0, 2L).…”

“…It is well-known ( [19]) that IBVP above has a unique solution if the functions y 0 , y 1 , g j , and f j are smooth enough. Our goal is to find the conditions on the system under consideration that guarantee the existence of controls f j (t) such that the ring goes to rest in a given time T and construct these controls.…”

On An Optimization Problem with Hyperbolic Partial Differential Equations

“…As for the control of initial conditions, Lions mentioned the control of the initial velocity of the system in detail but stated briefly the control of initial status of the system solving the . The notation of the functional spaces and their definitions are given in [9].…”