Variational Problems with Concentration

161 Pages·1999·5.389 MB·English

by  Martin Flucher (auth.)

Progress in Nonlinear Differential Equations
and Their Applications
Volume 36
Editor
Haim Brezis
Universite Pierre et Marie Curie
Paris
and
Rutgers University
New Brunswick, N.J.
Editorial Board
Antonio Ambrosetti, Scuola Internazionale Superiore di Studi Avanzati, Trieste
A. Bahri, Rutgers University, New Brunswick
Felix Browder, Rutgers University, New Brunswick
Luis Cafarelli, Institute for Advanced Study, Princeton
Lawrence C. Evans, University of California, Berkeley
Mariano Giaquinta, University of Pisa
David Kinderlehrer, Carnegie-Mellon University, Pittsburgh
Sergiu Klainerman, Princeton University
Robert Kohn, New York University
P.L. Lions, University of Paris IX
Jean Mahwin, Universite Catholique de Louvain
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
John Toland, University of Bath
Martin Flucher
Variational Problems
with Concentration
Springer Basel AG
Martin Flucher
SHERPA'XAG
Glutz-Blotzheimstr. 1
4503 Solothurn
Switzerland
1991 Mathematics Subject Classification 35J50; 35R35, 35J60, 35Q99
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Flucher, Martin:
Variational problems with concentration / Martin Flucher. -
Basel; Boston ; Berlin : Birkhäuser, 1999
(Progress in nonlinear differential equations and their applications ; Vol. 36)
ISBN 978-3-0348-9729-7 ISBN 978-3-0348-8687-1 (eBook)
DOI 10.1007/978-3-0348-8687-1
This work is subject to copyright. All rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, re-use of illustra
tions, broadcasting, reproduction on microfilms or in other ways, and storage in databanks.
For any kind of use whatsoever, permission from the copyright owner must be obtained.
© 1999 Springer Basel AG
Originally published by Birkhäuser Verlag, Basel, Switzerland in 1999
Softcover reprint of the hardcover 1st edition 1999
Printed on acid-free paper produced of chlorine-free pulp. TCF °°
ISBN 978-3-0348-9729-7
98765432 1
To my teacher Jurgen Moser
Contents
1 Introduction 1
2 P-Capacity . 13
3 Generalized Sobolev Inequality
3.1 Local generalized Sobolev inequality 18
3.2 Critical power integrand 20
3.3 Volume integrand . 20
3.4 Plasma integrand . . . . 21
4 Concentration Compactness Alternatives
4.1 CCA for critical power integrand 24
4.2 Generalized CCA . . . . . . . . 26
4.3 CCA for low energy extremals . 31
5 Compactness Criteria
5.1 Anisotropic Dirichlet energy . 35
5.2 Conformal metrics ..... . 37
6 Entire Extremals
6.1 Radial symmetry of entire extremals ...... . 44
6.2 Euler Lagrange equation (independent variable) . 45
6.3 Second order decay estimate for entire extremals 47
7 Concentration and Limit Shape of Low Energy Extremals
7.1 Concentration of low energy extremals . 52
7.2 Limit shape of low energy extremals .. 54
7.3 Exploiting the Euler Lagrange equation 59
8 Robin Functions
8.1 P-Robin function 63
8.2 Robin function for the Laplacian ..... 66
8.3 Conformal radius and Liouville's equation 67
vii
viii Contents
8.4 Computation of Robin function . . . . . 70
8.4.1 Boundary element method .... 70
8.4.2 Computation of conformal radius 73
8.4.3 Computation of harmonic centers . 76
8.5 Other Robin functions ...... . 76
8.5.1 Helmholtz harmonic radius 77
8.5.2 Biharmonic radius 78
9 P-Capacity of Small Sets. . . 81
10 P-Harmonic Transplantation 87
11 Concentration Points, Subconformal Case
11.1 Lower bound . . . . . . . . . . . . . 92
11.2 Identification of concentration points 93
12 Conformal Low Energy Limits
12.1 Concentration limit ... . 98
12.2 Conformal CCA .... . 100
12.3 Trudinger-Moser inequality 103
12.4 Concentration of low energy extremals 105
13 Applications
13.1 Optimal location of a small spherical conductor 109
13.2 Restpoints on an elastic membrane 111
13.3 Restpoints on an elastic plate . . 113
13.4 Location of concentration points 114
14 Bernoulli's Free-boundary Problem
14.1 Variational methods ..... . 119
14.2 Elliptic and hyperbolic solutions 121
14.3 Implicit Neumann scheme .... 126
14.4 Optimal shape of a small conductor 127
15 Vortex Motion
15.1 Planar hydrodynamics ........... . 131
15.2 Hydrodynamic Green's and Robin function 133
15.3 Point vortex model ....... . 137
15.4 Core energy method . . . . . . . 139
15.5 Motion of isolated point vortices 140
15.6 Motion of vortex clusters .... 142
15.7 Stability of vortex pairs . . . . . 145
15.8 Numerical approximation of vortex motion 147
Bibliography 151
Index .... 161
Chapter 1
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.
The corresponding Euler Lagrange equation is Bernoulli's free-boundary prob
lem with the constant Q playing the role of a Lagrange multiplier. In order to
transform this problem into a more standard variational problem we introduce
the volume integrand
(t < 1),
F(t)
(t ~ 1).
1
M. Flucher, Variational Problems with Concentration
© Birkhäuser Verlag 1999
2 Chapter 1. Introduction
u=O
-.D.u = 0
Figure 1.1: Solution of Bernoulli's free-boundary problem with equipotential
lines.
Solutions of Bernoulli's free-boundary problem are then constructed by maxi
mizing
In
F(u) I{u ~ 1}1
in a suitable space of functions with zero boundary data and fixed Dirichlet
In l'VuI
integral 2. The optimal insulation layer is finally obtained as a sublevel
set of the optimal potential 0 \ A = {O < u < I}. In terms of this example one
of the aims of the present monograph is to analyze the geometrical properties
of a small conductor A. We will characterize both its shape and location in a
form that is suitable for engineering purposes. In particular several numerical
schemes for the numerical computation of its asymptotic location will be devised
(Section 8.4). A detailed discussion of Bernoulli's free-boundary problem is given
in Chapter 14.
Plasma problem
Our second application arises in plasma physics (Section 3.4). The aim is to
determine the shape of a plasma ring in a tokamak. In its simplest version the
problem can be described as follows. Given the cross-section 0 of the tokomak,
find the plasma region A c 0 and the magnetic potential u such that:
-.D.u A(U - 1)+-1 in 0 \ 8A,
u 0 on 80,
u 1 on 8A,
Chapter 1. Introduction 3
-~(u-1)=A(u-1)
1
Figure 1.2: Solution of plasma problem on ball in ]R3 with q = 2.
u is 0 1 across 8A.
Figure 1.2 shows an example with q = 2. Variational solutions of this free
boundary problem can be obtained by maximizing
L
F(u)
with
(t ;::: 1),
F(t) { :' - 1)'
(t < 1)
among functions with zero boundary data and fixed Dirichlet integral. The re
sulting Euler equation is the above free-boundary problem involving a Lagrange
multiplier A. Again we are interested in the shape and location of thin plasma
rings.
General boundary value problem
More generally we consider nonlinear Dirichlet problems of the form
(1.1) f(u) in n,
u 0 on 8n
involving the p-Laplacian in the range 1 < p ::::; n where n denotes the space
dimension. Important special cases are p = 2 corresponding to the semilinear
equation
-/-l~U = f(u)
and the conformal case p = n. As to the viscosity pamm eter /-l we consider the
vanishing viscosity limit /-l -+ O.