Partial Differential Equations

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.

I. Elements of Spectral Theory.- §0. Introductory Remarks.- §1. Basic Definitions.- 1.0. Introductory Remarks.- 1.1. Fundamental Structure.- 1.2. Special Subsets.- 1.3. Operators.- 1.4. Functionals.- §2. The Spectrum of an Operator.- 2.0. Preliminary Remarks.- 2.1. Basic Definitions.- 2.2. Functions of an Operator.- 2.3. Connection Between the Spectra of an Operator and its Inverse.- §3. Special Classes of Operators.- 3.0. Preliminary Remarks.- 3.1. CC Operator s. Definition and Basic Properties.- 3.2. CC Operators. Fredholm-Riesz Theory.- 3.3. Selfadjoint CC Operators.- 3.4. Selfadjoint, Normal, and Unitary Operators.- 3.5. Some Additional Conventions.- 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.- 3.1. The Minimal Operator.- 3.2. Transposed Operations and Adjoint Operators.- 3.3. Existence of Proper Operators.- §4. Weak and Strong Extensions of Differential Operations.- 4.0. Preliminary Remarks.- 4.1. Basic Definitions.- §5. Averaging Operators.- 5.0. Preliminary Remarks.- 5.1. Averaging on the Line.- 5.2. Averaging in a Multidimensional Domain.- 5.3. Averaging and the Differentiation Operation.- 5.4. The Friedrichs Lemma.- §6. The Identity of Weak and Strong Extensions of Differential Operations.- 6.1. The Case of Mainly Constant Coefficients.- 6.2. The Case of Variable Coefficients.- 6.3. Some Examples.- 6.4. Equivalence of Weak and Strong Extensions as a Corollary of the Unique Solvability Problems.- 6.5. The Case of an Ordinary Differential Operator.- §7. W Spaces.- 7.0. Introductory Remarks.- 7.1. Weak and Strong Generalized Derivatives.- 7.2. Spaces Wn\ Wm, and Embedding Theorems.- III. Ordinary Differential Operators.- §0. Introductory Remarks.- §1. Description of Proper Operators for n = 1..- 1.1. Operators Generated by Cauchy Conditions.- 1.2. Description of Proper Operators.- §2. The Ordinary Differential Operator of the First Order.- §3. Birkhoff Theory.- §4. Supplementary Remarks.- 4.1. General Remarks.- 4.2. Supplementary Remarks Concerning Ordinary Differential Operations.- IV. Model Operators.- §0. Introductory Remarks.- §1. Tensor Products and Model Operators.- 1.1. Tensor Products of Hilbert Spaces.- 1.2. Model Operators.- §2. Operators on the n-Dimensional Torus.- 2.1. Definition of /7-Operators and their Basic Properties….- 2.2. Some Further Properties of 17-Operators.- 2.3. il-Operators Generated by Some Classical Differential Operations.- 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.- 1.0. Preparatory Remarks.- 1.1. Elementary Formulas.- 1.2. A General Method.- 1.3. The Cauchy Problem.- 1.4. Existence of Proper Operators.- 1.5. The Dirichlet Problem.- 1.6. Using the Standard Conditions.- 1.7. A Concluding Remark.- §2. Operator Equations of Higher Order (m 2).- 2.0. Preparatory Remarks.- 2.1. Binomial Equations.- 2.2. The General Operator Equation.- 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.- 3.0. Preparatory Remarks.- 3.1. Description of a Proper Operator by an Appropriate Choice of Basis.- 3.2. Existence of a Correctly Chosen Basis.- 3.3. The Final Result.- VIII. A Special Operational Calculus.- §0. Introductory Remarks.- §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.