no. 5
Un problème aux limites mixte des surfaces minimales avec une multiple projection plane et le dessin optimal des escaliers tournants
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1247-1270.

One admires rotational staircases in classical buildings since centuries. In particular, we are fascinated and inspired by the beautiful winding staircase (please, regard the picture below) in the center of the recently constructed University Library of the Brandenburgian Technical University at Cottbus by the bureau of architects Herzog & de Meuron from Basel. The sophisticated mathematician directly recognizes this staircase being a rotational minimal surface – namely the well-known helicoid – with a multiply covering projection onto the plane, solving a semi-free boundary value problem. We now ask the question, in which class of surfaces this helicoid is uniquely determined. Furthermore, we examine in how far the boundary values can be perturbed such that neighboring surfaces still exist. Both questions being affirmatively answered, we receive the stability of this boundary value problem. Finally, we investigate that our surface realizes a global minimum of area in the class of all parametric minimal surfaces solving an adequate mixed boundary value problem. Here we study one-to-one harmonic mappings onto the universal covering of the plane. This is achieved on the basis of our joint investigations with Professor Stefan Hildebrandt from the University of Bonn. Since H. Catalan was the first to classify the helicoid among ruled minimal surfaces and J. Plateau contributed, besides his inspiring experiments with soap bubbles, also his name to our central problem, I would like to present this treatise in the French language. During the construction of our University Library I got acquainted to the responsible architect for this project from the bureau Herzog & de Meuron, Frau Christine Binswanger and would like to dedicate this work to her with great respect. In her home city of Basel, classical Analysis could originally be developed by members of the Bernoulli family and Leonhard Euler.

DOI : 10.1016/j.anihpc.2010.06.002
Classification : 35J60, 53A10 
@article{AIHPC_2010__27_5_1247_0,
     author = {Sauvigny, Friedrich},
     title = {Un probl\`eme aux limites mixte des surfaces minimales avec une multiple projection plane et le dessin optimal des escaliers tournants},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1247--1270},
     publisher = {Elsevier},
     volume = {27},
     number = {5},
     year = {2010},
     doi = {10.1016/j.anihpc.2010.06.002},
     mrnumber = {2683759},
     zbl = {1210.00037},
     language = {fr},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.06.002/}
}
TY  - JOUR
AU  - Sauvigny, Friedrich
TI  - Un problème aux limites mixte des surfaces minimales avec une multiple projection plane et le dessin optimal des escaliers tournants
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 1247
EP  - 1270
VL  - 27
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2010.06.002/
DO  - 10.1016/j.anihpc.2010.06.002
LA  - fr
ID  - AIHPC_2010__27_5_1247_0
ER  - 
%0 Journal Article
%A Sauvigny, Friedrich
%T Un problème aux limites mixte des surfaces minimales avec une multiple projection plane et le dessin optimal des escaliers tournants
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 1247-1270
%V 27
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2010.06.002/
%R 10.1016/j.anihpc.2010.06.002
%G fr
%F AIHPC_2010__27_5_1247_0
Sauvigny, Friedrich. Un problème aux limites mixte des surfaces minimales avec une multiple projection plane et le dessin optimal des escaliers tournants. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1247-1270. doi : 10.1016/j.anihpc.2010.06.002. http://www.numdam.org/articles/10.1016/j.anihpc.2010.06.002/

[1] U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal Surfaces, Grundlehren der mathematischen Wissenschaften vol. 339, Springer-Verlag, Berlin (2010) | MR

[2] U. Dierkes, S. Hildebrandt, A. Tromba, Regularity of Minimal Surfaces, Grundlehren der mathematischen Wissenschaften vol. 400, Springer-Verlag, Berlin (2010) | MR

[3] U. Dierkes, S. Hildebrandt, A. Tromba, Global Analysis of Minimal Surfaces, Grundlehren der mathematischen Wissenschaften vol. 401, Springer-Verlag, Berlin (2010) | MR | Zbl

[4] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften vol. 224, Springer-Verlag, Berlin (1983) | MR | Zbl

[5] S. Hildebrandt, F. Sauvigny, Embeddedness and uniqueness of minimal surfaces solving a partially free boundary value problem, J. Reine Angew. Math. 422 (1991), 69-89 | EuDML | MR | Zbl

[6] S. Hildebrandt, F. Sauvigny, On one-to-one harmonic mappings and minimal surfaces, Nachr. Akad. Wiss. Göttingen, II Math.-Phys. Klasse 3 (1992), 73-93 | MR

[7] S. Hildebrandt, F. Sauvigny, Uniqueness of stable minimal surfaces with partially free boundaries, J. Math. Soc. Japan 47 no. 3 (1995), 423-440 | MR | Zbl

[8] F. Sauvigny, Partial Differential Equations. Part 1: Foundations and Integral Representations; Part 2: Functional Analytic Methods; With Consideration of Lectures by E. Heinz, Universitext, Springer-Verlag, Berlin (2006) | MR

Cité par Sources :