Direct Methods in the Theory of Elliptic Equations

• Jindřich Nečas
Book

Part of the Springer Monographs in Mathematics book series (SMM)

1. Front Matter
Pages i-xvi
2. Jindřich Nečas
Pages 1-47
3. Jindřich Nečas
Pages 49-114
4. Jindřich Nečas
Pages 115-195
5. Jindřich Nečas
Pages 197-246
6. Jindřich Nečas
Pages 247-279
7. Jindřich Nečas
Pages 281-317
8. Jindřich Nečas
Pages 319-346
9. Back Matter
Pages 347-372

Introduction

Nečas’ book Direct Methods in the Theory of Elliptic Equations, published 1967 in French, has become a standard reference for the mathematical theory of linear elliptic equations and systems. This English edition, translated by G. Tronel and A. Kufner, presents Nečas’ work essentially in the form it was published in 1967. It gives a timeless and in some sense definitive treatment of a number issues in variational methods for elliptic systems and higher order equations. The text is recommended to graduate students of partial differential equations, postdoctoral associates in Analysis, and scientists working with linear elliptic systems. In fact, any researcher using the theory of elliptic systems will benefit from having the book in his library.

The volume gives a self-contained presentation of the elliptic theory based on the "direct method", also known as the variational method. Due to its universality and close connections to numerical approximations, the variational method has become one of the most important approaches to the elliptic theory. The method does not rely on the maximum principle or other special properties of the scalar second order elliptic equations, and it is ideally suited for handling systems of equations of arbitrary order. The prototypical examples of equations covered by the theory are, in addition to the standard Laplace equation, Lame’s system of linear elasticity and the biharmonic equation (both with variable coefficients, of course). General ellipticity conditions are discussed and most of the natural boundary condition is covered. The necessary foundations of the function space theory are explained along the way, in an arguably optimal manner. The standard boundary regularity requirement on the domains is the Lipschitz continuity of the boundary, which "when going beyond the scalar equations of second order" turns out to be a very natural class. These choices reflect the author's opinion that the Lame system and the biharmonic equations are just as important as the Laplace equation, and that the class of the domains with the Lipschitz continuous boundary (as opposed to smooth domains) is the most natural class of domains to consider in connection with these equations and their applications.

Keywords

35J15, 35J45, 35J55 Sobolev space elliptic equation existence irregular domain regularity weak solution

Authors and affiliations

• Jindřich Nečas
• 1
• 2
1. 1.Charles UniversityPragueCzech Republic
2. 2.Northern Illinois UniversityDeKalbUSA

Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-10455-8
• Copyright Information Springer-Verlag Berlin Heidelberg 2012
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-642-10454-1
• Online ISBN 978-3-642-10455-8
• Series Print ISSN 1439-7382