Variational Problems with ConcentrationBirkhäuser, 6 gru 2012 - 163 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. |
Spis treści
1 | |
PCapacity of Small Sets | 9 |
Concentration Compactness Alternatives | 23 |
Compactness Criteria | 35 |
Entire Extremals | 43 |
Concentration and Limit Shape of Low Energy Extremals | 51 |
Vortex Motion | 97 |
Robin Functions | 136 |
151 | |
54 | 155 |
161 | |
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assume asymptotic Bandle Bernoulli's free-boundary problem best Sobolev constant biharmonic boundary element method compactness criterion computed concentration compactness alternative concentration limit concentration point conformal radius convergence convex critical power integrand defined denotes dimensions Dirichlet Dirichlet problem domain elliptic entire extremals estimate Euler Lagrange equation exterior ball condition extremal functions F(uɛ F(we Figure finite follows formula G₂ Green's function growth condition 1.3 harmonic center hydrodynamic hydrodynamic Green's function hyperbolic solutions isoperimetric inequality L¹(N Lagrange equation Lagrange multiplier Lemma lim sup linear Liouville's equation low energy extremals maximizing sequence method minimization motion nonlinearities obtain optimal p-capacity potential p-Green's p-harmonic radius p-Robin pcap Proof properties radial renormalized Robin function semilinear singularity small balls Sobolev inequality subconformal symmetrization Theorem 7.4 Trudinger-Moser inequality vanishes variational problem volume integrand vortices yields δυ ΘΩ