Invitation to Partial Differential Equations (Graduate Studies in Mathematics)

This book is based on notes from a beginning graduate course on partial differential equations. Prerequisites for using the book are a solid undergraduate course in real analysis. There are more than 100 exercises in the book. Some of them are just exercises, whereas others, even though they do require new ideas to solve them, provide additional important information about the subject. Cover Title page Foreword Preface Selected notational conventions Chapter 1. Linear differential operators 1.1. Definition and examples 1.2. The total and the principal symbols 1.3. Change of variables 1.4. The canonical form of second-order operators with constant coefficients 1.5. Characteristics. Ellipticity and hyperbolicity 1.6. Characteristics and the canonical form of second-order operators and second-order equations for n=2 1.7. The general solution of a homogeneous hyperbolic equation with constant coefficients for n=2 1.8. Appendix. Tangent and cotangent vectors 1.9. Problems Chapter 2. One-dimensional wave equation 2.1. Vibrating string equation 2.2. Unbounded string. The Cauchy problem. D’Alembert’s formula 2.3. A semibounded string. Reflection of waves from the end of the string 2.4. A bounded string. Standing waves. The Fourier method (separation of variables method) 2.5. Appendix. The calculus of variations and classical mechanics 2.6. Problems Chapter 3. The Sturm-Liouville problem 3.1. Formulation of the problem 3.2. Basic properties of eigenvalues and eigenfunctions 3.3. The short-wave asymptotics 3.4. The Green’s function and completeness of the system of eigenfunctions 3.5. Problems Chapter 4. Distributions 4.1. Motivation of the definition. Spaces of test functions 4.2. Spaces of distributions 4.3. Topology and convergence in the spaces of distributions 4.4. The support of a distribution 4.5. Differentiation of distributions and multiplication by a smooth function 4.6. A general notion of the transposed (adjoint) operator. Change of variables. Homogeneous distributions 4.7. Appendix. Minkowski inequality 4.8. Appendix. Completeness of distribution spaces 4.9. Problems Chapter 5. Convolution and Fourier transform 5.1. Convolution and direct product of regular functions 5.2. Direct product of distributions 5.3. Convolution of distributions 5.4. Other properties of convolution. Support and singular support of a convolution 5.5. Relation between smoothness of a fundamental solution and that of solutions of the homogeneous equation 5.6. Solutions with isolated singularities. A removable singularity theorem for harmonic functions 5.7. Estimates of derivatives of a solution of a hypoelliptic equation 5.8. Fourier transform of tempered distributions 5.9. Applying the Fourier transform to find fundamental solutions 5.10. Liouville’s theorem 5.11. Problems Chapter 6. Harmonic functions 6.1. Mean-value theorems for harmonic functions 6.2. The maximum principle 6.3. Dirichlet’s boundary value problem 6.4. Hadamard’s example 6.5. Green’s function for the Laplacian 6.6. Hölder regularity 6.7. Explicit formulas for Green’s functions 6.8. Problems Chapter 7. The heat equation 7.1. Physical meaning of the heat equation 7.2. Boundary value problems for the heat and Laplace equations 7.3. A proof that the limit function is harmonic 7.4. A solution of the Cauchy problem for the heat equation and Poisson’s integral 7.5. The fundamental solution for the heat operator. Duhamel’s formula 7.6. Estimates of derivatives of a solution of the heat equation 7.7. Holmgren’s principle. The uniqueness of solution of the Cauchy problem for the heat equation 7.8. A scheme for solving the first and second initial-boundary value problems by the Fourier method 7.9. Problems Chapter 8. Sobolev spaces. A generalized solution of Dirichlet’s problem 8.1. Spaces H^{s}(Ω) 8.2. Spaces \HH^{s}(Ω) 8.3. Dirichlet’s integral. The Friedrichs inequality 8.4. Dirichlet’s problem (generalized solutions) 8.5. Problems Chapter 9. The eigenvalues and eigenfunctions of the Laplace operator 9.1. Symmetric and selfadjoint operators in Hilbert space 9.2. The Friedrichs extension 9.3. Discreteness of spectrum for the Laplace operator in a bounded domain 9.4. Fundamental solution of the Helmholtz operator and the analyticity of eigenfunctions of the Laplace operator at the interior points. Bessel’s equation 9.5. Variational principle. The behavior of eigenvalues under variation of the domain. Estimates of eigenvalues 9.6. Problems Chapter 10. The wave equation 10.1. Physical problems leading to the wave equation 10.2. Plane, spherical, and cylindric waves 10.3. The wave equation as a Hamiltonian system 10.4. A spherical wave caused by an instant flash and a solution of the Cauchy problem for the three-dimensional wave equation 10.5. The fundamental solution for the three-dimensional wave operator and a solution of the nonhomogeneous wave equation 10.6. The two-dimensional wave equation (the descent method) 10.7. Problems Chapter 11. Properties of the potentials and their computation 11.1. Definitions of potentials 11.2. Functions smooth up to Γ from each side, and their derivatives 11.3. Jumps of potentials 11.4. Calculating potentials 11.5. Problems Chapter 12. Wave fronts and short-wave asymptotics for hyperbolic equations 12.1. Characteristics as surfaces of jumps 12.2. The Hamilton-Jacobi equation. Wave fronts, bicharacteristics, and rays 12.3. The characteristics of hyperbolic equations 12.4. Rapidly oscillating solutions. The eikonal equation and the transport equations 12.5. The Cauchy problem with rapidly oscillating initial data 12.6. Problems Chapter 13. Answers and hints. Solutions Bibliography Index Back Cover This book is based on notes from a beginning graduate course on partial differential equations. Prerequisites for using the book are a solid undergraduate course in real analysis. There are more than 100 exercises in the book. Some of them are just exercises, whereas others, even though they may require new ideas to solve them, provide additional important information about the subject. It is a great pleasure to see this book—written by a great master of the subject—finally in print. This treatment of a core part of mathematics and its applications offers the student both a solid foundation in basic calculations techniques in the subject, as well as a basic introduction to the more general machinery, e.g., distributions, Sobolev spaces, etc., which are such a key part of any modern treatment. As such this book is ideal for more advanced undergraduates as well as mathematically inclined students from engineering or the natural sciences. Shubin has a lovely intuitive writing style which provides a gentle introduction to this beautiful subject. Many good exercises (and solutions) are provided! —Rafe Mazzeo, Stanford University This text provides an excellent semester's introduction to classical and modern topics in linear PDE, suitable for students with a background in advanced calculus and Lebesgue integration. The author intersperses treatments of the Laplace, heat, and wave equations with developments of various functional analytic tools, particularly distribution theory and spectral theory, introducing key concepts while deftly avoiding heavy technicalities. —Michael Taylor, University of North Carolina, Chapel Hill