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Variational Problems with Concentration

  • Martin Flucher

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 36)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Martin Flucher
    Pages 1-11
  3. Martin Flucher
    Pages 13-15
  4. Martin Flucher
    Pages 17-22
  5. Martin Flucher
    Pages 23-33
  6. Martin Flucher
    Pages 35-42
  7. Martin Flucher
    Pages 43-50
  8. Martin Flucher
    Pages 63-79
  9. Martin Flucher
    Pages 81-86
  10. Martin Flucher
    Pages 87-90
  11. Martin Flucher
    Pages 97-107
  12. Martin Flucher
    Pages 109-115
  13. Martin Flucher
    Pages 117-129
  14. Martin Flucher
    Pages 131-149
  15. Back Matter
    Pages 151-163

About this book

Introduction

To start with we describe two applications of the theory to be developed in this monograph: Bernoulli's free-boundary problem and the plasma problem. Bernoulli's free-boundary problem This problem arises in electrostatics, fluid dynamics, optimal insulation, and electro chemistry. In electrostatic terms the task is to design an annular con­ denser consisting of a prescribed conducting surface 80. and an unknown conduc­ tor A such that the electric field 'Vu is constant in magnitude on the surface 8A of the second conductor (Figure 1.1). This leads to the following free-boundary problem for the electric potential u. -~u 0 in 0. \A, u 0 on 80., u 1 on 8A, 8u Q on 8A. 811 The unknowns are the free boundary 8A and the potential u. In optimal in­ sulation problems the domain 0. \ A represents the insulation layer. Given the exterior boundary 80. the problem is to design an insulating layer 0. \ A of given volume which minimizes the heat or current leakage from A to the environment ]R.n \ n. The heat leakage per unit time is the capacity of the set A with respect to n. Thus we seek to minimize the capacity among all sets A c 0. of equal volume.

Keywords

Euler–Lagrange equation Robin functions Variational calculus compactness differential equation extrema numerical methods partial differential equation

Authors and affiliations

  • Martin Flucher
    • 1
  1. 1.SHERPA’X AGSolothurnSwitzerland

Bibliographic information