1st Edition

Variational Techniques for Elliptic Partial Differential Equations Theoretical Tools and Advanced Applications

    514 Pages
    by CRC Press

    514 Pages 20 B/W Illustrations
    by CRC Press

    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

    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


    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