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AuthorDezin, Aleksei A. author
TitlePartial Differential Equations [electronic resource] : An Introduction to a General Theory of Linear Boundary Value Problems / by Aleksei A. Dezin
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1987
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Descript XII, 165 p. online resource


Let me begin by explaining the meaning of the title of this book. In essence, the book studies boundary value problems for linear partial differยญ 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


I. Elements of Spectral Theory -- ยง0. Introductory Remarks -- ยง1. Basic Definitions -- ยง2. The Spectrum of an Operator -- ยง3. Special Classes of Operators -- II. Function Spaces and Operators Generated by Differentiation -- ยง0. Introductory Remarks -- ยง1. The Space IH(F) -- ยง2. Differential Operations and the Maximal Operator -- ยง3. The Minimal Operator and Proper Operators -- ยง4. Weak and Strong Extensions of Differential Operations -- ยง5. Averaging Operators -- ยง6. The Identity of Weak and Strong Extensions of Differential Operations -- ยง7. W Spaces -- III. Ordinary Differential Operators -- ยง0. Introductory Remarks -- ยง1. Description of Proper Operators for n = 1. -- ยง2. The Ordinary Differential Operator of the First Order -- ยง3. Birkhoff Theory -- ยง4. Supplementary Remarks -- IV. Model Operators -- ยง0. Introductory Remarks -- ยง1. Tensor Products and Model Operators -- ยง2. Operators on the n-Dimensional Torus -- V. First-Order Operator Equations -- ยง0. Introductory Remarks -- ยง1. The Operator Dtโ{128}{148}A; The Spectrum -- ยง2. The Operator Dt2013A; Special Boundary Conditions -- ยง3. The Operator Dtโ{128}{148}k\Classification -- ยง4. Operators not Solvable for Dt -- ยง5. 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 -- ยง0. Introductory Remarks -- ยง1. Second-Order Operator Equations -- ยง2. Operator Equations of Higher Order (m 2) -- VII. General Existence Theorems for Proper Operators.. -- ยง0. Introductory Remarks -- ยง1. Lemma on Restriction of a Domain -- ยง2. Existence Theorem for a Proper Operator -- 3. Description of Proper Operators in a Parallelepiped -- VIII. A Special Operational Calculus -- ยง0. Introductory Remarks -- ยง1. Construction of the Operational Calculus -- ยง2. Some Examples -- ยง3. The Necessity for Restrictions on the Resolvent -- Concluding Remarks -- Appendix 1. On Some Systems of Equations Containing a Small Parameter -- ยง1. Formulation of the Problem -- ยง2. Truncation of the System -- ยง3. The Complete System -- Appendix 2. Further Developments -- References -- Index of Symbols

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