Variational problems with concentration

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 conƯ denser consisting of a prescribed conducting surface 80. and an unknown conducƯ 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 inƯ 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.

1 Introduction -- 2 P-Capacity -- 3 Generalized Sobolev Inequality -- 3.1 Local generalized Sobolev inequality -- 3.2 Critical power integrand -- 3.3 Volume integrand -- 3.4 Plasma integrand -- 4 Concentration Compactness Alternatives -- 4.1 CCA for critical power integrand -- 4.2 Generalized CCA -- 4.3 CCA for low energy extremals -- 5 Compactness Criteria -- 5.1 Anisotropic Dirichlet energy -- 5.2 Conformai metrics -- 6 Entire Extremals -- 6.1 Radial symmetry of entire extremals -- 6.2 Euler Lagrange equation (independent variable) -- 6.3 Second order decay estimate for entire extremals -- 7 Concentration and Limit Shape of Low Energy Extremals -- 7.1 Concentration of low energy extremals -- 7.2 Limit shape of low energy extremals -- 7.3 Exploiting the Euler Lagrange equation -- 8 Robin Functions -- 8.1 P-Robin function -- 8.2 Robin function for the Laplacian -- 8.3 Conformai radius and Liouville's equation -- 8.4 Computation of Robin function -- 8.5 Other Robin functions -- 9 P-Capacity of Small Sets -- 10 P-Harmonic Transplantation -- 11 Concentration Points, Subconformai Case -- 11.1 Lower bound -- 11.2 Identification of concentration points -- 12 Conformai Low Energy Limits …