Variational problems with concentration /
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 correspo...
| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Basel ; Boston :
Birkhauser,
©1999
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| Series: | Progress in nonlinear differential equations and their applications ;
v. 36 |
| Subjects: |
| Summary: | 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 readers benefit, the presentation is self-contained and new techniques are explained in detail. Bernoullis 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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| Physical Description: | 1 online resource (viii, 163 pages) : illustrations |
| Format: | Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002. |
| Bibliography: | Includes bibliographical references (pages 151-160) and index |
| ISBN: | 303488687X 3034897294 9783034886871 9783034897297 |