This self-contained textbook offers an elementary introduction to partial differential equations (PDEs), primarily focusing on linear equations, but also providing a perspective on nonlinear equations, through Hamilton--Jacobi equations, elliptic equations with measurable coefficients and DeGiorgi classes. The exposition is complemented by examples, problems, and solutions that enhance understanding and explore related directions. Large parts of this revised second edition have been streamlined and rewritten to incorporate years of classroom feedback, correct misprints, and improve clarity. The work can serve as a text for advanced undergraduates and graduate students in mathematics, physics, engineering, and the natural sciences, as well as an excellent reference for applied mathematicians and mathematical physicists. The newly added three last chapters, on first order non-linear PDEs (Chapter 8), quasilinear elliptic equations with measurable coefficients (Chapter 9) and DeGiorgi classes (Chapter 10), point to issues and directions at the forefront of current investigations. Reviews of the first edition: The author's intent is to present an elementary introduction to PDEs... In contrast to other elementary textbooks on PDEs . . . much more material is presented on the three basic equations: Laplace's equation, the heat and wave equations. . . . The presentation is clear and well organized. . . . The text is complemented by numerous exercises and hints to proofs. ---Mathematical Reviews This is a well-written, self-contained, elementary introduction to linear, partial differential equations. ---Zentralblatt MATH
CONTENT
Preface to the Second Edition -- Preface to the First Edition -- Preliminaries -- Quasi-Linear Equations -- The LaPlace Equation -- Boundary Value Problems by Double Layer Potentials -- Integral Equations and Eigenvalue Problems -- The Heat Equation -- The Wave Equation -- Quasi-Linear Equations of First Order -- Non-Linear Equations of First Order -- References -- Index
SUBJECT
Mathematics
Functional equations
Fourier analysis
Integral equations
Differential equations
partial
Mathematical optimization
Mathematical physics
Mathematics
Partial Differential Equations
Fourier Analysis
Difference and Functional Equations
Integral Equations
Calculus of Variations and Optimal Control; Optimization
