This book presents a clear path from calculus to classical potential theory and beyond with the aim of moving the reader into a fertile area of mathematical research as quickly as possible. The first half of the book develops the subject matter from first principles using only calculus. The second half comprises more advanced material for those with a senior undergraduate or beginning graduate course in real analysis. For specialized regions, solutions of Laplace’s equation are constructed having prescribed normal derivatives on the flat portion of the boundary and prescribed values on the remaining portion of the boundary. By means of transformations known as diffeomorphisms, these solutionsare morphed into local solutions on regions with curved boundaries. The Perron-Weiner-Brelot method is then used to construct global solutions for ellipticPDEs involving a mixture of prescribed values of a boundary differential operator on part of the boundary and prescribed values on the remainder of the boundary. Written by the author of Introduction to Potential Theory, this is a new and modern textbook that introduces all the important concepts of classical potential theory Equips readers for further study in elliptic partial differential equations, axiomatic potential theory, and the interplay between probability theory and potential theory INDICE: Preface.- Preliminaries.- 1. Laplace's Equation.- 2. The DirichletProblem.- 3. Green Functions.- 4. Negligible Sets.- 5. Dirichlet Problem for Unbounded Regions.- 6. Energy.- 7. Interpolation and Monotonicity.- 8. Newtonian Potential.- 9. Elliptic Operators.- 10. Apriori Bounds.- 11. Oblique Derivative Problem.- References.- Index.- Notation.
- ISBN: 978-1-84882-318-1
- Editorial: Springer
- Encuadernacion: Rústica
- Páginas: 455
- Fecha Publicación: 01/06/2009
- Nº Volúmenes: 1
- Idioma: Inglés