1st Edition
Variational Techniques for Elliptic Partial Differential Equations Theoretical Tools and Advanced Applications
I Fundamentals
1 Distributions
2 The homogeneous Dirichlet problem
3 Lipschitz transformations and Lipschitz domains
4 The nonhomogeneous Dirichlet problem
5 Nonsymmetric and complex problems
6 Neumann boundary conditions
7 Poincare inequalities and Neumann problems
8 Compact perturbations of coercive problems
9 Eigenvalues of elliptic operators
II Extensions and Applications
10 Mixed problems
11 Advanced mixed problems
12 Nonlinear problems
13 Fourier representation of Sobolev spaces
14 Layer potentials
15 A collection of elliptic problems
16 Curl spaces and Maxwell's equations
17 Elliptic equations on boundaries
A Review material
B Glossary
Biography
Francisco-Javier Sayas is a Professor of Mathematical Sciences at the University of Delaware. He has published over one hundred research articles in refereed journals, and is the author of Retarded Potentials and Time Domain Boundary Integral Equations.
Thomas S. Brown is a lecturer in Computational and Applied Mathematics at Rice University. He received his PhD in Mathematics from the University of Delaware in 2018, under the supervision of Francisco-Javier Sayas. His expertise lies in the theoretical and numerical study of elastic wave propagation in piezoelectric media with applications to control problems.
Matthew E. Hassell is a Systems Engineer at Lockheed Martin. He received his PhD in Applied Mathematics from the University of Delaware in 2016, under the supervision of Francisco-Javier Sayas, working on convolution quadrature techniques for problems in wave propagation and scattering by non-homogeneous media as well as viscous flow around obstacles.
"This book proposes a modern and practical approach to the classical subject of elliptic partial differential equations. It provides the correct functional framework for proving the existence of weak solutions in a broad selection of model problems, in anticipation of their numerical approximation (with finite element and boundary element methods).
Assuming little beyond basic undergraduate mathematics, the book covers, step by step, an amount of topics that goes beyond most textbooks at this level. The first part (Chapters 1-9) is a self-contained and meticulously written text on linear elliptic boundary-value problems for graduate students. The remaining chapters tread in less explored territories with topics that cannot be found in many other textbooks. They include, for example, the study of Maxwell's equations with a detailed characterisation of the tangential trace spaces on Lipschitz domains, the Stokes-Darcy problem, dependence with respect to coefficients and elliptic problems on manifolds.
Each chapter ends with final comments written in an attractive and personal style with judicious references to specialised literature. In addition, there is a good selection of problems that offers students the opportunity to further expand their knowledge.
The richness of material, the clear exposition and distilled writing will appeal to students with interest in partial differential equations and their numerical analysis as well as to professional mathematicians. Even experienced instructors may benefit from the thoughtful approach of the book and from its original insight on the subject."
-Salim Meddahi, Universidad de Oviedo
