| Variational Problems with Concentration 1999 Edition Contributor(s): Bach, Martin F. (Author) |
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ISBN: 3764361360 ISBN-13: 9783764361365 Publisher: Birkhauser
Binding Type: Hardcover - See All Available Formats & Editions Published: July 1999 Annotation: 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.Bernoulli's 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. Click for more in this series: Progress in Nonlinear Differential Equations and Their Applications |
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| Additional Information |
| BISAC Categories: - Mathematics | Differential Equations - Partial - Medical - Mathematics | Number Systems |
| Dewey: 515.353 |
| LCCN: 99038073 |
| Series: Progress in Nonlinear Differential Equations and Their Applications |
| Physical Information: 0.44" H x 6.14" W x 9.21" L (0.95 lbs) 163 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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 con- denser consisting of a prescribed conducting surface 80. and an unknown conduc- 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 in- 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. |
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