Sayas F.J., Brown T.S., Hassell M.E. Variational Techniques for Elliptic Partial Differential Equations: Theoretical Tools and Advanced Applications

Boca Raton: CRC Press, 2019. — 514 p. — ISBN: 1138580880.Variational Techniques for Elliptic Partial Differential Equations, intended for graduate students studying applied math. analysis, and/or numerical analysis, provides the necessary tools to understand the structure and solvability of elliptic partial differential equations. Beginning with the necessary definitions and theorems from distribution theory, the book gradually builds the functional analytic framework for studying elliptic PDE using variational formulations. Rather than introducing all of the prerequisites in the first chapters, it is the introduction of new problems which motivates the development of the associated analytical tools. In this way the student who is encountering this material for the first time will be aware of exactly what theory is needed, and for which problems.Features:A detailed and rigorous development of the theory of Sobolev spaces on Lipschitz domains, including the trace operator and the normal component of vector fields.
An integration of functional analysis concepts involving Hilbert spaces and the problems which can be solved with these concepts, rather than separating the two.
Introduction to the analytical tools needed for physical problems of interest like time-harmonic waves, Stokes and Darcy flow, surface differential equations, Maxwell cavity problems, etc.
A variety of problems which serve to reinforce and expand upon the material in each chapter, including applications in fluid and solid mechanics.Authors.
Fundamentals
Distributions.
The homogeneous Dirichlet problem.
Lipschitz transformations and Lipschitz domains.
The nonhomogeneous Dirichlet problem.
Nonsymmetric and complex problems.
Neumann boundary conditions.
Poincare inequalities and Neumann problems.
Compact perturbations of coercive problems.
Eigenvalues of elliptic operators.
Extensions and Applications
Mixed problems.
Advanced mixed problems.
Nonlinear problems.
Fourier representation of Sobolev spaces.
Layer potentials.
A collection of elliptic problems.
Curl spaces and Maxwell’s equations.
Elliptic equations on boundaries.
Appendix A Review material
Appendix B Glossary

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