Asymptotic analysis for surfaces with large constant mean curvature and free boundaries
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 1, pp. 109-129.

We prove that simply connected H-surfaces with bounded area and free boundary in a domain necessarily concentrate at a critical point of the mean curvature of the boundary of this domain.

@article{AIHPC_2012__29_1_109_0,
     author = {Laurain, Paul},
     title = {Asymptotic analysis for surfaces with large constant mean curvature and free boundaries},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {109--129},
     publisher = {Elsevier},
     volume = {29},
     number = {1},
     year = {2012},
     doi = {10.1016/j.anihpc.2011.09.004},
     zbl = {1242.53009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.004/}
}
TY  - JOUR
AU  - Laurain, Paul
TI  - Asymptotic analysis for surfaces with large constant mean curvature and free boundaries
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
SP  - 109
EP  - 129
VL  - 29
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.004/
DO  - 10.1016/j.anihpc.2011.09.004
LA  - en
ID  - AIHPC_2012__29_1_109_0
ER  - 
%0 Journal Article
%A Laurain, Paul
%T Asymptotic analysis for surfaces with large constant mean curvature and free boundaries
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 109-129
%V 29
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.004/
%R 10.1016/j.anihpc.2011.09.004
%G en
%F AIHPC_2012__29_1_109_0
Laurain, Paul. Asymptotic analysis for surfaces with large constant mean curvature and free boundaries. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 1, pp. 109-129. doi : 10.1016/j.anihpc.2011.09.004. http://www.numdam.org/articles/10.1016/j.anihpc.2011.09.004/

[1] Sami Baraket, Estimations of the best constant involving the L norm in Wenteʼs inequality, Ann. Fac. Sci. Toulouse Math. (6) 5 no. 3 (1996), 373-385 | EuDML | Numdam | Zbl

[2] H. Brezis, J.-M. Coron, Convergence of solutions of H-systems or how to blow bubbles, Arch. Ration. Mech. Anal. 89 no. 1 (1985), 21-56 | Zbl

[3] Haïm Brezis, Jean-Michel Coron, Multiple solutions of H-systems and Rellichʼs conjecture, Comm. Pure Appl. Math. 37 no. 2 (1984), 149-187 | Zbl

[4] Isaac Chavel, Riemannian Geometry: A Modern Introduction, Cambridge Studies in Advanced Mathematics vol. 98, Cambridge University Press, Cambridge (2006) | Zbl

[5] Tobias H. Colding, William P. Minicozzi, Minimal Surfaces, Courant Lecture Notes in Mathematics vol. 4, New York University, Courant Institute of Mathematical Sciences, New York (1999) | Zbl

[6] Ulrich Dierkes, Stefan Hildebrandt, Albrecht Küster, Ortwin Wohlrab, Minimal Surfaces. II: Boundary Regularity, Grundlehren der Mathematischen Wissenschaften vol. 296, Springer-Verlag, Berlin (1992) | Zbl

[7] Olivier Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc. 130 no. 8 (2002), 2351-2361 | Zbl

[8] Olivier Druet, Emmanuel Hebey, Frédéric Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry, Mathematical Notes vol. 45, Princeton University Press, Princeton, NJ (2004) | Zbl

[9] Mouhamed Moustapha Fall, Embedded disc-type surfaces with large constant mean curvature and free boundaries, preprint SISSA, 2007.

[10] Mouhamed Moustapha Fall, Area-minimizing regions with small volume in Riemannian manifolds with boundary, Pacific J. Math. 244 no. 2 (2010), 235-260 | Zbl

[11] Mouhamed Moustapha Fall, Fethi Mahmoudi, Hypersurfaces with free boundary and large constant mean curvature: Concentration along submanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 7 no. 3 (2008), 407-446 | EuDML | Numdam | Zbl

[12] Yuxin Ge, Estimations of the best constant involving the L 2 norm in Wenteʼs inequality and compact H-surfaces in Euclidean space, ESAIM Control Optim. Calc. Var. 3 (1998), 263-300 | EuDML | Numdam | Zbl

[13] David Gilbarg, Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin (2001) | Zbl

[14] Heinz Hopf, Differential Geometry in the Large, Lecture Notes in Mathematics vol. 1000, Springer-Verlag, Berlin (1983) | Zbl

[15] Katsuei Kenmotsu, Surfaces with Constant Mean Curvature, Translations of Mathematical Monographs vol. 221, American Mathematical Society, Providence, RI (2003) | Zbl

[16] Nicholas J. Korevaar, Rob Kusner, Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 no. 2 (1989), 465-503 | Zbl

[17] Paul Laurain, Concentration of CMC surfaces in a Riemannian manifold, I.M.R.N., in press.

[18] Frank Morgan, Geometric Measure Theory: A Beginnerʼs Guide, Elsevier/Academic Press, Amsterdam (2009)

[19] Manuel Ritoré, Antonio Ros, Some updates on isoperimetric problems, Math. Intelligencer 24 no. 3 (2002), 9-14

[20] Antonio Ros, Enaldo Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 no. 1 (1995), 19-33 | Zbl

[21] Michael Struwe, Plateauʼs Problem and the Calculus of Variations, Mathematical Notes vol. 35, Princeton University Press, Princeton, NJ (1988) | Zbl

[22] Peter Topping, The optimal constant in Wenteʼs L estimate, Comment. Math. Helv. 72 no. 2 (1997), 316-328 | Zbl

[23] Henry C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318-344 | Zbl

[24] Henry C. Wente, Large solutions to the volume constrained Plateau problem, Arch. Ration. Mech. Anal. 75 no. 1 (1980/81), 59-77 | Zbl

[25] Ye Rugang, Foliation by constant mean curvature spheres, Pacific J. Math. 147 no. 2 (1991), 381-396 | Zbl

Cité par Sources :