Bandle / Vergara Caffarelli / Berestycki Elliptic and Parabolic Problems

A Special Tribute to the Work of Haim Brezis
1. Auflage 2006
ISBN: 978-3-7643-7384-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 63, 470 Seiten

Reihe: Progress in Nonlinear Differential Equations and Their Applications

Haim Brezis has made significant contributions in the fields of partial differential equations and functional analysis, and this volume collects contributions by his former students and collaborators in honor of his 60th anniversary at a conference in Gaeta. It presents new developments in the theory of partial differential equations with emphasis on elliptic and parabolic problems.

Autoren/Hrsg.

Bandle, Catherine

Berestycki, Henri

Brighi, Bernhard

Vergara Caffarelli, Giorgio

Brillard, Alain

Chipot, Michel

Coron, Jean-Michel

Sbordone, Carlo

Shafrir, Itai

Coro, Jean-Michel

Valente, Vanda

Inhaltsverzeichnis

1;Contents;6
2;Preface;10
3;One-Layer Free Boundary Problems with Two Free Boundaries;11
3.1;1. Introduction;11
3.2;2. Notation, deffnitions, preliminary results;13
3.3;3. Existence, uniqueness, monotonicity, and continuous dependence of solutions;14
3.4;4. Successive approximation of solutions;17
3.5;5. Concluding remarks;21
3.6;References;21
4;New Numerical Solutions for the Brezis-Nirenberg Problem on Sn;23
4.1;1. Introduction;23
4.2;2. Numerical strategy;24
4.3;3. Numerical results;27
4.4;References;30
5;On some Boundary Value Problems for Incompressible Viscous Flows with Shear Dependent Viscosity;33
5.1;1. Introduction;33
5.2;2. The Stokes stationary problem;35
5.3;3. The evolution problem;39
5.4;4. Appendix;41
5.5;References;42
6;Radiative Heat Transfer in Silicon Purification;43
6.1;1. Introduction;43
6.2;2. The physical problem;43
6.3;3. The mathematical model;44
6.4;4. Numerical solution: time and space discretization;48
6.5;5. Numerical results;50
6.6;References;52
7;A Decay Result for a Quasilinear Parabolic System;53
7.1;1. Introduction;53
7.2;2. Preliminaries;55
7.3;References;59
8;The Shape of Charged Drops: Symmetrybreaking Bifurcations and Numerical Results;61
8.1;1. Introduction;61
8.2;2. The bifurcation result;63
8.3;3. Numerical method;66
8.4;4. Numerical results and discussion;67
8.5;References;68
9;On the One-dimensional Parabolic Obstacle Problem with Variable Coeffcients;69
9.1;1. Introduction;69
9.2;2. The notion of blow-up;71
9.3;3. A monotonicity formula for energy;72
9.4;4. A Liouville’s theorem and consequences;73
9.5;5. A monotonicity formula for singular points;74
9.6;References;75
10;Hardy Potentials and Quasi-linear Elliptic Problems Having Natural Growth Terms;77
10.1;1. Introduction;77
10.2;2. Right-hand side like;78
10.3;3. ;86
10.4;4. Lower order terms and weak summability of the data;87
10.5;5. Degenerate coercivity of the principal part;89
10.6;References;91
11;Recent Advances on Similarity Solutions Arising During Free Convection;93
11.1;1. Introduction;93
11.2;2. The case of prescribed heat;94
11.3;3. The case of prescribed heat flux;99
11.4;4. Asymptotic behavior of the unbounded solutions;100
11.5;References;100
12;Rellich Relations for Mixed Boundary Elliptic Problems;103
12.1;Introduction;103
12.2;1. Rellich relation for the Laplace equation;104
12.3;2. Rellich relation for the Lame system;106
12.4;3. Sketch of the proof of Theorem 4;110
12.5;References;112
13;Lyapunov - type Inequalities and Applications to PDE;113
13.1;1. Introduction;113
13.2;2. Lyapunov - type inequalities for the linear problem;114
13.3;3. The nonlinear problem;119
13.4;References;120
14;Gaeta 2004. Elliptic Resonant Problems with a Periodic Nonlinearity;121
14.1;1. Introduction;121
14.2;2. The variational approach: nondegeneracy;122
14.3;References;127
15;Harnack Inequality for p-Laplacians on Metric Fractals;129
15.1;1. Introduction;129
15.2;2. Metric fractals;130
15.3;3. Harnack inequality;132
15.4;4. Examples;133
15.5;References;135
16;Wave Propagation in Discrete Media;137
16.1;1. Introduction;137
16.2;2. Stationary wave fronts;139
16.3;3. Travelling wave fronts;140
16.4;4. Depinning transitions;141
16.5;5. Models with inertia;142
16.6;References;143
17;A Solution of the Heat Equation with a Continuum of Decay Rates;145
17.1;References;148
18;Finite Volume Scheme for Semiconductor Energy- transport Model;149
18.1;1. Introduction;149
18.2;2. Formulation of the model;150
18.3;3. The finite volume scheme;151
18.4;4. Simulation of an n+ nn+ ballistic diode;152
18.5;References;155
19;Asymptotic Behavior of Nonlinear Parabolic Problems with Periodic Data;157
19.1;1. Introduction;157
19.2;2. The main result;158
19.3;References;166
20;Geodesic Computations for Fast and Accurate Surface Remeshing and Parameterization;167
20.1;1. Introduction;168
20.2;2. Geodesic Remeshing;169
20.3;3. Fast Geodesic Parameterization;173
20.4;4. Results and Discussion;176
20.5;5. Conclusion;180
20.6;References;180
21;On the Newton Body Type Problems;183
21.1;References;186
22;Some Open Problems on Water Tank Control Systems;189
22.1;1. Modelling equations of a water tank control system;189
22.2;2. Controllability: results and open problems;191
22.3;3. A toy model;194
22.4;References;198
23;Hölder Estimates for Solutions to a Singular Nonlinear Neumann Problem;199
23.1;1. Introduction;199
23.2;2. Preliminaries;201
23.3;3. Proof of Theorem;203
23.4;4. Proof of Theorem;213
23.5;References;214
24;Asymptotic Analysis of the Neumann Problem for the Ukawa Equation in a Thick Multi- structure of Type 3: 2: 2;217
24.1;1. Background and objective of the study;217
24.2;2. The convergence theorem;220
24.3;3. Asymptotic approximation;222
24.4;References;224
25;On the Häim Brezis Pioneering Contributions on the Location of Free Boundaries;227
25.1;1. Introduction;227
25.2;2. The support of the solution of a variational inequality in fluid mechanics;228
25.3;3. The support of the solution of semilinear (multivalued or sublinear) second order equations;232
25.4;4. Compact support properties and the abstract theory of monotone operators;236
25.5;5. Special acknowledgements;240
25.6;References;240
26;Fractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization;245
26.1;1. Introduction;245
26.2;2. Existence of a global solution;246
26.3;3. Regularity and uniqueness of the solution;248
26.4;4. Vanishing regularization;249
26.5;References;252
27;Stationary and Self-similar Solutions for Coagulation and Fragmentation Equations;253
27.1;1. Introduction;253
27.2;2. Stationary and self similar solutions;255
27.3;3. The new results;258
27.4;4. Main ingredients of the proofs;261
27.5;5. Asymptotic behavior of the solutions to the fragmentation equation;265
27.6;References;266
28;Orlicz Capacities and Applications to PDEs and Sobolev Mappings;269
28.1;1. A nonexistence result;269
28.2;2. A capacitary estimate;273
28.3;References;275
29;Energy Forms on Non Self-similar Fractals;277
29.1;1. Introduction;277
29.2;2. Energy and Lagrangian on self-similar fractals;278
29.3;3. Energy form on non self-similar fractals;281
29.4;References;286
30;Measure Data and Numerical Schemes for Elliptic Problems;289
30.1;1. Introduction;289
30.2;2. A model example;291
30.3;3. Convection-diffusion equations;295
30.4;References;300
31;Brezis-Nirenberg Problem and Coron Problem for Polyharmonic Operators;301
31.1;1. Introduction;301
31.2;2. Existence of positive solutions for general domains;303
31.3;3. Existence of positive solutions for some perforated domains;304
31.4;References;306
32;Local and Global Properties of Solutions of a Nonlinear Boundary Layer Equation;309
32.1;1. Introduction;309
32.2;2. Multiple solutions and large behavior;311
32.3;3. Behavior of singular solutions at the blowing-up point;314
32.4;References;316
33;Mathematical Models of Aggregation: The Role of Explicit Solutions;319
33.1;1. Introduction;319
33.2;2. The polymerization equations;321
33.3;3. Kinetic coagulation equations: mathematical theory;324
33.4;4. Reversible coagulation: the impact of fragmentation;325
33.5;5. Related problems and open questions;326
33.6;References;327
34;Metastable Behavior of Premixed Gas Flames;329
34.1;1. Introduction;329
34.2;2. Physical model;330
34.3;3. Numerical simulations and results;331
34.4;4. One-dimensional case and rectangular domains;333
34.5;References;336
35;Recent Progress on Boundary Blow-up;339
35.1;1. Introduction;340
35.2;2. Fuchsian Reduction;341
35.3;3. Two types of Fuchsian elliptic operators;348
35.4;4. Outline of proof of Theorem;349
35.5;References;349
36;Maximum Principle for Bounded Solutions of the Telegraph Equation: The Case of High Dimensions;353
36.1;1. Introduction;353
36.2;2. Maximum principles for the telegraph equation when;355
36.3;3. Obstruction to a maximum principle for;359
36.4;References;361
37;Kolmogorov Equations in Physics and in Finance;363
37.1;1. Introduction;363
37.2;2. Constant coefficients Kolmogorov equations;367
37.3;3. Kolmogorov equations with regular coeffcients;368
37.4;4. Kolmogorov equations with measurable coefficients;370
37.5;References;372
38;Harnack Inequalities and Gaussian Estimates for a Class of Hypoelliptic Operators;375
38.1;1. Introduction;375
38.2;2. Heat operators on Carnot groups;378
38.3;3. Kolmogorov type operators;379
38.4;4. Operators on linked groups;380
38.5;5. Outline of the proof;381
38.6;References;383
39;How to Construct Good Measures;385
39.1;1. Introduction;385
39.2;2. Construction of the Cantor set F associated to the subsequence ;387
39.3;3. Potential generated by the uniform measure;388
39.4;4. Proofs of Theorems 2 and 3 ;391
39.5;5. Proof of Theorem 1;395
39.6;References;397
40;Bifurcation and Asymptotics for Elliptic Problems with Singular Nonlinearity;399
40.1;1. Motivation and previous results;399
40.2;2. A singular problem with sublinear nonlinearity;401
40.3;3. Bifurcation and asymptotics for a singular elliptic equation with convection term;406
40.4;4. An elliptic problem with strong singular nonlinearity and convection term;408
40.5;References;411
41;A Model for Hysteresis in Mechanics Using Local Minimizers of Young Measures;413
41.1;1. Introduction;413
41.2;2. Evolution of Young measures;415
41.3;3. One-dimensional systems;417
41.4;4. Applications;420
41.5;References;422
42;The Precise Lp-theory of Elliptic Equations in the Plane;425
42.1;1. Introduction;425
42.2;2. The non divergence equation;427
42.3;3. The adjoint to non divergence equation;428
42.4;References;430
43;Essential Spectrum and Noncontrollability of Membrane Shells;433
43.1;1. Introduction;433
43.2;2. The nonlinear problem;434
43.3;3. The linearized equations;436
43.4;References;440
44;The Porous Medium Equation. New Contractivity Results;443
44.1;1. Introduction;443
44.2;2. Progress in the PME;444
44.3;3. Contractivity and the Porous Medium Equation;445
44.4;4. The Wasserstein metrics;446
44.5;5. Contractivity in One Space dimension;448
44.6;6. The contractivity question in several space dimensions;451
44.7;7. Open problems and comments;457
44.8;References;458
45;Large Solutions of Elliptic Equations with Strong Absorption;463
45.1;Introduction;463
45.2;1. Proof of the main results;465
45.3;2. Applications;468
45.4;References;473
46;Relaxation in Presence of Pointwise Gradient Constraints;475
46.1;References;480

Leseproben

Radiative Heat Transfer in Silicon Purification (P.33)

A. Bermudez, R. Leira, M.C. Müniz and F. Pena

Abstract.

We present a numerical model describing the thermal behavior of a silicon purification process which takes place into a so-called casting ladle. We consider, simultaneously, the phase change in the silicon and a nonlinear non-local boundary condition arising from the Stefan-Boltzmann radiation condition at the enclosure surfaces within the ladle. We also propose a numerical approximation using a finite element method. An iterative algorithm and numerical results are presented.

1. Introduction

In many engineering applications involving high-temperature processes numerical simulation provides an insight into the radiative analysis of these complex systems and it promotes improvements of several process optimization (see [3, 5]). The motivation of this work is to compute the numerical solution of the problem addressed in [6] applied to a silicon purification process – see [1]. We consider, simultaneously, the phase change in the silicon and the non-local boundary condition arising from the Stefan-Boltzmann radiation condition at the enclosure surfaces within the ladle.

The outline of this paper is as follows. In Section 2 the physical problem is introduced. In Section 3 using the axisymmetry of the domain, we formulate the mathematical problem in a two-dimensional domain by means of cylindrical coordinates. Section 4 is devoted to introduce space and time discretization of the aforementioned problem and to present an iterative algorithm. Finally, in Section 5, several numerical results are shown.

2. The physical problem

Metallurgical grade silicon (MG-Si) is obtained from a silicon oxide in electrical submerged arc furnaces. A technique of MG-Si puri.cation is to melt it and to induce its directional solidification. This method of removing impurities is based on the fact that most impurities tend to remain in a molten region rather than re-solidify.

This purification process is taking place into a casting ladle which consists of a finite axisymmetric cylinder containing a cylindrical enclosure. After the casting ladle being electrically heated, its lid is open and molten silicon is poured into its inner cavity keeping a gap between the top of the silicon and the upper part of the inner ladle surface where several heating elements are located. The objective is now to push upwards the metal impurities by means of inducing its one-directional solidification switching on the heating elements and then keeping molten the top of the silicon ingot.

In doing so, the solid silicon grows gradually upwards into the liquid and the metallic impurities are segregated into the melt region during solidification, thus, at the end of the process most of impurities are concentrated at the top of the silicon ingot. Radiation heat transfer is considered at the inner cavity and materials of the enclosure are assumed to be opaque (see [4]), therefore radiation may be treated as a surface phenomenon.

Moreover, we assume both that the walls of the cylindrical enclosure behave as black surfaces and that the medium within the enclosure is radiatively nonparticipating so that it has no effect on the radiation transfer between inner surfaces.