Elliptic Partial Differential Equations from an Elementary Viewpoint

This is a textbook that covers several selected topics in the theory of elliptic partial differential equations which can be used in an advanced undergraduate or graduate course.

The book considers many important issues such as existence, regularity, qualitative properties, and all the classical topics useful in the wide world of partial differential equations. It also includes applications with interesting examples.

The structure of the book is flexible enough to allow different chapters to be taught independently.

The book is friendly, welcoming, and written for a newcomer to the subject.

It is essentially self-contained, making it easy to read, and all the concepts are fully explained from scratch, combining intuition and rigor, and therefore it can also be read independently by students, with limited or no supervision.

Contents:

• What is the Laplacian?
• The Laplace Operator and Harmonic Functions
• Upper Semicontinuity and Subharmonicity
• Equations in Nondivergence Form: C^2.α-Regularity Theory
• Equations in Nondivergence Form: W^2.p-Regularity Theory
• The Dirichlet Problem in the Light of Capacity Theory
• Some Interesting Problems Arising from the Poisson Equation
• The Moving Plane Method
• Local Existence Theory in the Real Analytic Setting
• Appendix A: An Interesting Example
• Appendix B: An Application to Physical Geodesy

Readership: Undergraduate and graduate students in applied mathematics courses, lecturers and researchers.

Key Features:

• It is accessible to third-year undergraduate students, but of high interest to advanced students and possibly professional researchers
• All the topics covered are of a classical flavor, but the exposition and the organization of text is quite original
• Difficult topics are addressed with a style that is as self-contained as possible, rigorous as well as intuitive, and approachable without major prerequisites
• Only the basics of analysis are necessary. References to useful results and theorems are offered for less advanced readers
• All the sections and chapters can be mixed and combined according to the preferences of the lecturer or of the reader to organize a bespoke course on elliptic PDEs