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.
Erscheinungsdatum: 02.11.2012

lgrsnf/K:\!genesis\0day\kolxoz\80\M_Mathematics\MC_Calculus\MCde_Differential equations\Flucher M. Variational problems with concentration (PNDE036, Birkhauser, 1999)(ISBN 9783034897297)(600dpi)(T)(161s)_MCde_.djvu

lgli/M_Mathematics/MC_Calculus/MCde_Differential equations/Flucher M. Variational problems with concentration (PNDE036, Birkhauser, 1999)(ISBN 9783034897297)(600dpi)(T)(161s)_MCde_.djvu

nexusstc/Variational Problems with Concentration/413b1811f31137977e9a33912dcd59a6.djvu

scihub/10.1007/978-3-0348-8687-1.pdf

zlib/Mathematics/Martin F. Bach/Variational Problems with Concentration_3372308.djvu

Martin F. Bach

Birkhäuser Basel : Imprint : Birkhäuser

Springer Nature Switzerland AG

Springer Basel AG

Birkhauser Verlag

Progress in nonlinear differential equations and their applications, v. 36, Basel, ©1999

Place of publication not identified, 1999

Springer Nature, Basel, 2012

Switzerland, Switzerland

3, 20121206

1, 2012

kolxoz -- 80

lg2130604

{"edition":"softcover reprint of the original 1st ed. 1999","isbns":["303488687X","3034897294","9783034886871","9783034897297"],"last_page":163,"publisher":"Birkhäuser Basel","series":"Progress in nonlinear differential equations and their applications 36"}

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.

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 conƯ denser consisting of a prescribed conducting surface 80. and an unknown conducƯ 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 inƯ 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

Front Matter....Pages i-viii
Introduction....Pages 1-11
P-Capacity....Pages 13-15
Generalized Sobolev Inequality....Pages 17-22
Concentration Compactness Alternatives....Pages 23-33
Compactness Criteria....Pages 35-42
Entire Extremals....Pages 43-50
Concentration and Limit Shape of Low Energy Extremals....Pages 51-62
Robin Functions....Pages 63-79
P-Capacity of Small Sets....Pages 81-86
P-Harmonic Transplantation....Pages 87-90
Identification of Concentration Points, Subconformal Case....Pages 91-95
Conformal Low Energy Limits....Pages 97-107
Applications....Pages 109-115
Bernoulli’s Free-boundary Problem....Pages 117-129
Vortex Motion in Two Dimensional Hydrodynamics....Pages 131-149
Back Matter....Pages 151-163

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.

2017-10-15