Partial Differential Equations: An Accessible Route Through Theory and Applications (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 169)

This text on partial differential equations is intended for readers who want to understand the theoretical underpinnings of modern PDEs in settings that are important for the applications without using extensive analytic tools required by most advanced texts. The assumed mathematical background is at the level of multivariable calculus and basic metric space material, but the latter is recalled as relevant as the text progresses. The key goal of this book is to be mathematically complete without overwhelming the reader, and to develop PDE theory in a manner that reflects how researchers would think about the material. A concrete example is that distribution theory and the concept of weak solutions are introduced early because while these ideas take some time for the students to get used to, they are fundamentally easy and, on the other hand, play a central role in the field. Then, Hilbert spaces that are quite important in the later development are introduced via completions which give essentially all the features one wants without the overhead of measure theory. There is additional material provided for readers who would like to learn more than the core material, and there are numerous exercises to help solidify one's understanding. Readership The text should be suitable for advanced undergraduates or for beginning graduate students including those in engineering or the sciences and also Professors, graduate students, and others interested in teaching and learning partial differential equations. Preface Chapter 1 Introduction 1. Preliminaries and notation 2. Partial differential equations Additional material: More on normed vector spaces and metric spaces Problems Chapter 2 Where do PDE come from? 1. An example: Maxwell’s equations 2. Euler-Lagrange equations Problems Chapter 3 First order scalar semilinear equations Additional material: More on ODE and the inverse function theorem Problems Chapter 4 First order scalar quasilinear equations Problems Chapter 5 Distributions and weak derivatives Additional material: The space L 1 Problems Chapter 6 Second order constant coefficient PDE: Types and d’Alembert’s solution of the wave equation 1. Classification of second order PDE Problems Chapter 7 Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle 1. Properties of solutions of the wave equation: Propagation phenomena 2. Energy conservation for the wave equation 3. The maximum principle for Laplace’s equation and the heat equation 4. Energy for Laplace’s equation and the heat equation Problems Chapter 8 The Fourier transform: Basic properties, the inversion formula and the heat equation 1. The definition and the basics 2. The inversion formula 3. The heat equation and convolutions 4. S y stem s o f P D E 5. Integral transforms Additional material: A heat kernel proof of the Fourier inversion formula Problems Chapter 9 The Fourier transform:Tempered distributions, the wave equation and Laplace’s equation 1. Tempered distributions 2. The Fourier transform of tempered distributions 3. The wave equation and the Fourier transform 4. More on tempered distributions Problems Chapter 10 PDE and boundaries 1. The wave equation on a half space 2. The heat equation on a half space 3. More complex geometries 4. Boundaries and properties of solutions 5. PDE on intervals and cubes Problems Chapter 11 Duhamel’s principle 1. The inhomogeneous heat equation 2. The inhomogeneous wave equation Problems Chapter 12 Separation of variables 1. The general method 2. Interval geometries 3. Circular geometries Problems Chapter 13 Inner product spaces, symmetric operators, or thogonality 1. The basics of inner product spaces 2. Symmetric operators 3. Completeness of orthogonal sets and of the inner product space Problems Chapter 14 Convergence of the Fourier series and the Poisson formula on disks 1. Notions of convergence 2. Uniform convergence of the Fourier transform 3. What does the Fourier series converge to? 4. The Dirichlet problem on the disk Additional material: The Dirichlet kernel Problems Chapter 15 Bessel functions 1. The definition of Bessel functions 2. The zeros of Bessel functions 3. Higher dimensions Problems Chapter 16 The method of stationary phase Problems Chapter 17 Solvability via duality 1. The general method 2. An example: Laplace’s equation 3. Inner product spaces and solvability Problems Chapter 18 Variational problems 1. The finite dimensional problem 2. The infinite dimensional minimization Problems Bibliography Index Cover -- Title page -- Contents -- Preface -- Chapter 1. Introduction -- Chapter 2. Where do PDE come from? -- Chapter 3. First order scalar semilinear equations -- Chapter 4. First order scalar quasilinear equations -- Chapter 5. Distributions and weak derivatives -- Chapter 6. Second order constant coefficient PDE: Types and d'Alembert's solution of the wave equation -- Chapter 7. Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle -- Chapter 8. The Fourier transform: Basic properties, the inversion formula and the heat equation -- Chapter 9. The Fourier transform: Tempered distributions, the wave equation and Laplace's equation -- Chapter 10. PDE and boundaries -- Chapter 11. Duhamel's principle -- Chapter 12. Separation of variables -- Chapter 13. Inner product spaces, symmetric operators, orthogonality -- Chapter 14. Convergence of the Fourier series and the Poisson formula on disks -- Chapter 15. Bessel functions -- Chapter 16. The method of stationary phase -- Chapter 17. Solvability via duality -- Chapter 18. Variational problems -- Bibliography -- Index -- Back Cover 1. Introduction -- 2. Where Do Pde Come From? -- 3. First Order Scalar Semilinear Equations -- 4. First Order Scalar Quasilinear Equations -- 5. Distributions And Weak Derivatives -- 6. Second Order Constant Coefficient Pde: Types And D'alembert's Solution Of The Wave Equation -- 7. Properties Of Solutions Of Second Order Pde: Propagation, Energy Estimates And The Maximum Principle -- 8. The Fourier Transform: Basic Properties, The Inversion Formula And The Heat Equation -- 9. The Fourier Transform: Tempered Distributions, The Wave Equation And Laplace's Equation -- 10. Pde And Boundaries -- 11. Duhamel's Principle -- 12. Separation Of Variables -- 13. Inner Product Spaces, Symmetric Operators, Orthogonality -- 14. Convergence Of The Fourier Series And The Poisson Formula On Disks -- 15. Bessel Functions -- 16. The Method Of Stationary Phase -- 17. Solvability Via Duality -- 18. Variational Problems. Andras Vasy. Includes Bibliographical References And Index. Fundamentos teóricos de las ecuaciones diferenciales parciales modernas en entornos que son importantes para las aplicaciones sin necesidad de utilizar herramientas analíticas extensas requeridas por la mayoría de los libros de texto avanzados