Lectures on partial differential equation

Preface to the Second Russian Edition 1.The General Theory for One First-Order Equation Literature 2.The General Theory for One First-Order Equation (Continued) Literature 3.Huygens' Principle in the Theory of Wave Propagation 4.1.The Vibrating String (d'Alembert's Method) 4.1.The General Solution 4.2.Boundary-Value Problems and the Cauchy Problem 4.3.The Cauchy Problem for an Infinite String. d'Alembert's Formula 4.4.The Semi-Infinite String 4.5.The Finite String Resonance 4.6.The Fourier Method 5.The Fourier Method (for the Vibrating String) 5.1.Solution of the Problem in the Space of Trigonometric Polynomials 5.2.A Digression 5.3.Formulas for Solving the Problem of Section 5.1 5.4.The General Case 5.5.Fourier Series 5.6.Convergence of Fourier Series 5.7.Gibbs' Phenomenon 6.The Theory of Oscillations.The Variational Principle.adro 41 Literature 7.The Theory of Oscillations.The Variational Principle (Continued) 8.Properties of Harmonic Functions 8.1.Consequences of the Mean-Value Theorem 8.2.The Mean-Value Theorem in the Multidimensional Case 9.The Fundamental Solution for the Laplacian.Potentials 9.1.Examples and Properties 9.2.A Digression.The Principle of Superposition 9.3.Appendix.An Estimate of the Single-Layer Potential 10.The Double-Layer Potential 10.1.Properties of the Double-Layer Potential 11.Spherical Functions.Maxwell's Theorem.The Removable Singularities Theorem 12.Boundary-Value Problems for Laplace's Equation.Theory of Linear Equations and Systems 12.1.Four Boundary-Value Problems for Laplace's Equation 12.2.Existence and Uniqueness of Solutions 12.3.Linear Partial Differential Equations and Their Symbols A.The Topological Content of Maxwell's Theorem on the Multifeld Representation of Spherical Functions A.1.The Basic Spaces and Groups A.2.Some Theorems of Real Algebraic Geometry A.3.From Algebraic Geometry to Spherical Functions A.4.Explicit Formulasa A.5.Maxwell's Theorem and CP2/conj¡ÖS4 A.6.The History of Maxwell's Theorem Literature B.Problems B.1.Material from the Seminars B.2.Written Examination Problems