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Variational Methods for Boundary Value Problems for Systems of Elliptic Equations

M. A. Lavrent'ev
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
, on

Dover's Phoenix series reprints books that do not have the sales potential to be "normal" Dover books but that might still be of some interest to a smaller circle of readers. This one reprints a 1965 translation of a book first published in Russian in 1962. In a brief review published in the American Mathematical Monthly in 1965, A. Friedman described the book as an attempt 

…to present a unified approach to the study of elliptic systems consisting of two equations, two dependent and two independent variables. This approach is based upon first conceiving the solution of a problem as a minimum in some set-up, then taking a minimizaing sequence blelonging to a compact set and concluding, finally, that a convergent subsequence converges to a minimum because otherwise the limit could be modified to yield a smaller value.

He also points out that "the book is somewhat loosely written; the proofs are often sketchy or even partly omitted, although the theorems are stated precisely."



1. Variational principles
2. Sufficient conditions
3. Generalizations
Chapter 1. Variational principles of the theory of conformal mapping
  1.1 The principles of Lindelöf and Montel
    1.1.1 The case of the circle
    1.1.2 Mapping on to a strip
  1.2 Mechanical interpretation
  1.3 Quantitative estimates
Chapter 2. Behaviour of a conformal transformation on the boundary
  2.1 Derivatives at the boundary
  2.2 Narrow strips
  2.3 Behaviour of the extension at points of maximum inclination and extreme curvature
Chapter 3. Hydrodynamic applications
  3.1 Stream line flow
  3.2 Generalizations
  3.3 Stream line flow with detachment
  3.4 Wave motions of a fluid
  3.5 The linear theory of waves
  3.6 Rayleigh waves
  3.7 The exact theory
  3.8 Generalizations
    3.8.1 Motion of a fluid over a submarine trench
    3.8.2 Motion over a bottom with a ridge
    3.8.3 Spillway with singularities
Chapter 4. Quasi-conformal mappings
  4.1 The concept of the quasi-conformal map
  4.2 Derivative systems
  4.3 Strong ellipticity
Chapter 5. Linear systems
  5.1 Transformations with bounded distortion
    5.1.1 Equi-graded continuity
    5.1.2 Almost conformal mappings
  5.2 The simplest class of linear systems
    5.2.1 Invariance with respect to conformal mappings
    5.2.2. Stability of conformal mappings
    5.2.3 Condition of smoothness of a transformation
    5.2.4 Application to arbitrary linear systems
    5.2.5 Existence theorem
Chapter 6. The simplest classes of non-linear systems
  6.1 Maximum principle
  6.2 The principle of Schwarz-Lindelöf
  6.3 Quantitative estimates
  6.4 Inductive proof of Lindelöf's principle
  6.5 The existence theorem
  6.6 Generalizations
  6.7 Hydrodynamic applications
  References; Index