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.


  • 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

part Part I|1 pages


chapter 1|24 pages


chapter 2|20 pages

The homogeneous Dirichlet problem

chapter 4|24 pages

The nonhomogeneous Dirichlet problem

chapter 5|18 pages

Nonsymmetric and complex problems

chapter 6|24 pages

Neumann boundary conditions

chapter 8|28 pages

Compact perturbations of coercive problems

chapter 9|29 pages

Eigenvalues of elliptic operators

part Part II|1 pages

Extensions and Applications

chapter 10|44 pages

Mixed problems

chapter 11|24 pages

Advanced mixed problems

chapter 12|18 pages

Nonlinear problems

chapter 13|32 pages

Fourier representation of Sobolev spaces

chapter 14|36 pages

Layer potentials

chapter 15|46 pages

A collection of elliptic problems

chapter 16|44 pages

Curl spaces and Maxwell's equations

chapter 17|11 pages

Elliptic equations on boundaries