Variational Problems with Concentration - Martin F. Bach

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Résumé :
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Sommaire:

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.- 8.4.1 Boundary element method.- 8.4.2 Computation of conformai radius.- 8.4.3 Computation of harmonic centers.- 8.5 Other Robin functions.- 8.5.1 Helmholtz harmonic radius.- 8.5.2 Biharmonic radius.- 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.- 12.1 Concentration limit.- 12.2 Conformai CCA.- 12.3 Trudinger-Moser inequality.- 12.4 Concentration of low energy extremals.- 13 Applications.- 13.1 Optimal location of a small spherical conductor.- 13.2 Restpoints on an elastic membrane.- 13.3 Restpoints on an elastic plate.- 13.4 Location of concentration points.- 14 Bernoulli's Free-boundary Problem.- 14.1 Variational methods.- 14.2 Elliptic and hyperbolic solutions.- 14.3 Implicit Neumann scheme.- 14.4 Optimal shape of a small conductor.- 15 Vortex Motion.- 15.1 Planar hydrodynamics.- 15.2 Hydrodynamic Green's and Robin function.- 15.3 Point vortex model.- 15.4 Core energy method.- 15.5 Motion of isolated point vortices.- 15.6 Motion of vortex clusters.- 15.7 Stability of vortex pairs.- 15.8 Numerical approximation of vortex motion.