Variational Problems With Concentration

Przednia okładka
Springer Science & Business Media, 1999 - 163
The subject of this research monograph is semilinear Dirichlet problems and similar equations involving the p-Laplacian. Solutions are constructed by a constraint variational method. The major new contribution is a detailed analysis of low-energy solutions. In PDE terms the low-energy limit corresponds to the well-known vanishing viscosity limit. First it is shown that in the low-energy limit the Dirichlet energy concentrates at a single point in the domain. This behaviour is typical of a large class of nonlinearities known as zero mass case. Moreover, the concentration point can be identified in geometrical terms. This fact is essential for flux minimization problems. Finally, the asymptotic behaviour of low-energy solutions in the vicinity of the concentration point is analyzed on a microscopic scale. The sound analysis of the zero mass case is novel and complementary to the majority of research articles dealing with the positive mass case. It illustrates the power of a purely variational approach where PDE methods run into technical difficulties. To the readersâ benefit, the presentation is self-contained and new techniques are explained in detail. Bernoulliâs free-boundary problem and the plasma problem are the principal applications to which the theory is applied. The author derives several numerical methods approximating the concentration point and the free boundary. These methods have been implemented and tested by a co-worker. The corresponding plots are highlights of this book.
 

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Introduction
1
PCapacity
13
Generalized Sobolev Inequality
17
31 Local generalized Sobolev inequality
18
32 Critical power integrand
20
34 Plasma integrand
21
Concentration Compactness Alternatives
23
41 Concentration compactness alternative for critical power integrand
24
PHarmonic Transplantation
85
Identification of Concentration Points Subconformal Case
89
111 Lower bound
90
112 Upper bound and identification of concentration points
91
Conformal Low Energy Limits
95
122 Conformal concentration compactness alternative
98
123 Extremal functions for the TrudingerMoser inequality
101
124 Concentration of conformal low energy extremals
103

42 Generalized concentration compactness alternative
26
43 Concentration compactness alternative for low energy extremals
31
Compactness Criteria
35
52 Extremal functions for Sobolevs inequality with conformal metrics
37
Entire Extremals
43
61 Radial symmetry of entire extremals
44
62 Euler Lagrange equation variation of the independent variable
45
63 Second order decay estimate for entire extremals
47
Concentration and Limit Shape of Low Energy Extremals
51
71 Concentration of low energy extremals
52
72 Limit shape of low energy extremals
54
73 Exploiting the Euler Lagrange equation
59
Robin Functions
63
82 Robin function for the Laplacian
64
83 Conformal radius and Liouvilles equation
65
84 Computation of Robin function and harmonic centers
68
842 Computation of conformal radius
71
843 Computation of harmonic centers
74
851 Helmholtz harmonic radius
75
852 Biharmonic radius
76
PCapacity of Small Sets
79
Applications
107
132 Restpoints on an elastic membrane
109
133 Restpoints on an elastic plate
111
134 Location of concentration points in a semilinear Dirichlet problem
112
Bernoullis Freeboundary Problem
115
141 Variational methods for Bernoullis freeboundary problem
117
142 Nondegenerate elliptic hyperbolic solutions
119
143 Implicit Neumann scheme
124
144 Optimal shape of a small conductor
125
Vortex Motion in Two Dimensional Hydrodynamics
129
152 Hydrodynamic Greens and Robin function
131
153 Point vortex model
135
154 Core energy method
137
155 Motion of isolated point vortices
138
156 Motion of vortex clusters
140
157 Stability of vortex pairs
143
158 Numerical approximation of vortex motion
145
Bibliography
149
Index
159
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