Semilinear Elliptic Equations for Beginners

1. Front Matter

   Pages I-X

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2. Introduction and Basic Results

   • Marino Badiale, Enrico Serra

   Pages 1-37

3. Minimization Techniques: Compact Problems

   • Marino Badiale, Enrico Serra

   Pages 39-96

4. Minimization Techniques: Lack of Compactness

   • Marino Badiale, Enrico Serra

   Pages 97-143

5. Introduction to Minimax Methods

   • Marino Badiale, Enrico Serra

   Pages 145-189

6. Back Matter

   Pages 191-199

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Semilinear elliptic equations are of fundamental importance for the study of geometry, physics, mechanics, engineering and life sciences. The variational approach to these equations has experienced spectacular success in recent years, reaching a high level of complexity and refinement, with a multitude of applications. Additionally, some of the simplest variational methods are evolving as classical tools in the field of nonlinear differential equations. This book is an introduction to variational methods and their applications to semilinear elliptic problems. Providing a comprehensive overview on the subject, this book will support both student and teacher engaged in a first course in nonlinear elliptic equations. The material is introduced gradually, and in some cases redundancy is added to stress the fundamental steps in theory-building. Topics include differential calculus for functionals, linear theory, and existence theorems by minimization techniques and min-max procedures. Requiring a basic knowledge of Analysis, Functional Analysis and the most common function spaces, such as Lebesgue and Sobolev spaces, this book will be of primary use to graduate students based in the field of nonlinear partial differential equations. It will also serve as valuable reading for final year undergraduates seeking to learn about basic working tools from variational methods and the management of certain types of nonlinear problems.

• Boundary value problems
• Critical points
• Elliptic equations
• Minimax methods
• Minimization
• Variational methods
• partial differential equations

From the reviews:

“An excellent text suitable for a graduate student or a very well-prepared undergraduate. It can easily support either a regular class or an independent study format. … The authors have done a good job of avoiding esoteric technicalities, so … that a good student could learn the notions above while studying this text. The first chapter outlines much of this material and provides references for those who need further study. … the text is on variational methods applied to semilinear elliptic boundary value problems.” (Stephen B. Robinson, Mathematical Reviews, Issue 2012 f)

“This book is a valuable reference book for specialists in the field and an excellent graduate text giving an overview of the literature on solutions of semilinear elliptic equations. … the book should be strongly recommended to anyone, either graduate student or researcher, who is interested in variational methods and their applications to partial differential equations of elliptic type .” (Vicenţiu D. Rădulescu, Zentralblatt MATH, Vol. 1214, 2011)

• Dipartimento di Matematica, Università di Torino, Torino, Italy

  Marino Badiale, Enrico Serra

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