Partial Differential Equations: An Introduction to a General Theory of Linear Boundary Value Problems(Springer Series in Soviet Mathematics)

Let me begin by explaining the meaning of the title of this book. In essence, the book studies boundary value problems for linear partial differ- ential equations in a finite domain in n-dimensional Euclidean space. The problem that is investigated is the question of the dependence of the nature of the solvability of a given equation on the way in which the boundary conditions are chosen, i.e. on the supplementary requirements which the solution is to satisfy on specified parts of the boundary. The branch of mathematical analysis dealing with the study of boundary value problems for partial differential equations is often called mathematical physics. Classical courses in this subject usually consider quite restricted classes of equations, for which the problems have an immediate physical context, or generalizations of such problems. With the expanding domain of application of mathematical methods at the present time, there often arise problems connected with the study of partial differential equations that do not belong to any of the classical types. The elucidation of the correct formulation of these problems and the study of the specific properties of the solutions of similar equations are closely related to the study of questions of a general nature.

Table of Contents:
I. Elements of Spectral Theory.- §0. Introductory Remarks.- §1. Basic Definitions.- §2. The Spectrum of an Operator.- §3. Special Classes of Operators.- II. Function Spaces and Operators Generated by Differentiation.- §0. Introductory Remarks.- §1. The Space IH(F).- §2. Differential Operations and the Maximal Operator.- §3. The Minimal Operator and Proper Operators.- §4. Weak and Strong Extensions of Differential Operations.- §5. Averaging Operators.- §6. The Identity of Weak and Strong Extensions of Differential Operations.- §7. W Spaces.- III. Ordinary Differential Operators.- §0. Introductory Remarks.- §1. Description of Proper Operators for n = 1..- §2. The Ordinary Differential Operator of the First Order.- §3. Birkhoff Theory.- §4. Supplementary Remarks.- IV. Model Operators.- §0. Introductory Remarks.- §1. Tensor Products and Model Operators.- §2. Operators on the n-Dimensional Torus.- V. First-Order Operator Equations.- §0. Introductory Remarks.- §1. The Operator Dt—A; The Spectrum.- §2. The Operator Dt2013A; Special Boundary Conditions.- §3. The Operator Dt—k\Classification.- §4. Operators not Solvable for Dt.- §5. Differential Properties of the Solutions of Operator Equations, and Related Questions.- 6. Some Operators with Variable Coefficients in the Principal Part.- 7. Concluding Remarks.- VI. Operator Equations of Higher Order.- §0. Introductory Remarks.- §1. Second-Order Operator Equations.- §2. Operator Equations of Higher Order (m 2).- VII. General Existence Theorems for Proper Operators...- §0. Introductory Remarks.- §1. Lemma on Restriction of a Domain.- §2. Existence Theorem for a Proper Operator.- 3. Description of Proper Operators in a Parallelepiped.- VIII. A Special Operational Calculus.- §0. IntroductoryRemarks.- §1. Construction of the Operational Calculus.- §2. Some Examples.- §3. The Necessity for Restrictions on the Resolvent.- Concluding Remarks.- Appendix 1. On Some Systems of Equations Containing a Small Parameter.- §1. Formulation of the Problem.- §2. Truncation of the System.- §3. The Complete System.- Appendix 2. Further Developments.- References.- Index of Symbols.