Variational Problems with Concentration

ISBN-13: 9783034897297 / Angielski / Miękka / 2012 / 163 str.

Martin Flucher

Opis

This self-contained research monograph focuses on semilinear Dirichlet problems and similar equations involving the p-Laplacian. The author explains new techniques in detail, and derives several numerical methods approximating the concentration point and the free boundary. The corresponding plots are highlights of this book.

Recenzje

"This material is very rich, because various alternatives are possible. The book provides a good systematic overview of this difficult topic."

--Zentralblatt Math

Spis treści

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.