Abstract

We consider the minimization problem min⁡Ω∈X(Λ2−Λ∞)(Ω),\begin{aligned}\min_{\Omega\in X}\left(\Lambda_2-\Lambda_\infty\right)(\Omega),\end{aligned} where Λ2(Ω)\Lambda_2(\Omega) and Λ∞(Ω)\Lambda_\infty(\Omega) are the (square root of the) first eigenvalue of the Laplacian and the first eigenvalue of the ∞−\infty-Laplacian respectively. XX is the class of convex domains with prescribed diameter. We prove existence of a solution, and we provide several geometrical properties of minimizers

Author information

Contact details are reproduced from the original publication and may be historical.

Marino Belloni

Dip. di Matematica, Università di Parma, Viale G. P. Usberti 53/A, 43100 Parma, Italy

marino.belloni@unipr.it

Edouard Oudet

Laboratoire de Mathematiques, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-du-Lac, France

edouard.oudet@univ-savoie.fr

References

  1. G. Aronsson: Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967) 551–561.https://doi.org/10.1007/BF02591928Google Scholar
  2. G. Aronsson, M. G. Crandall, P. Juutinen: A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., New Ser. 41(4) (2004) 439–505.https://doi.org/10.1090/S0273-0979-04-01035-3Google Scholar
  3. M. Belloni, A. Wagner: The ∞ eigenvalue problem from a variational point of view, preprint (2006).Google Scholar
  4. D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems, Progress in Nonlinear Differential Equations and their Applications 65, Birkhäuser, Basel (2005).https://doi.org/10.1007/b137163Google Scholar
  5. M. G. Crandall, H. Ishii, P. L. Lions: User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., New Ser. 27(1) (1992) 1–67.https://doi.org/10.1090/S0273-0979-1992-00266-5Google Scholar
  6. A. Henrot: Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel (2006)https://doi.org/10.1007/3-7643-7706-2Google Scholar
  7. A. Henrot, M. Pierre: Variation et Optimisation de Formes, Mathématiques et Applications 48, Springer, Berlin (2005).https://doi.org/10.1007/3-540-37689-5Google Scholar
  8. J. Hersch: Sur la fréquence fondamentale d’une membrane vibrante: évaluations par défaut et principe de maximum, Z. Angew. Math. Phys. 11 (1960) 387–413.https://doi.org/10.1007/BF01604498Google Scholar
  9. P. Juutinen, P. Lindqvist, J. Manfredi: The ∞-eigenvalue problem, Arch. Ration. Mech. Anal. 148(2) (1999) 89–105.https://doi.org/10.1007/s002050050157Google Scholar
  10. B. Kawohl: Rearrangement and Convexity of Level Sets in PDE, Lecture Notes in Mathematics 1150, Springer, Berlin (1985).https://doi.org/10.1007/BFb0075060Google Scholar
  11. T. Lachand-Robert, É. Oudet: Bodies of constant width in arbitrary dimension, Math. Nachr. 280 (2007) 740–750.https://doi.org/10.1002/mana.200510512Google Scholar
  12. P. Lindqvist: On a nonlinear eigenvalue problem, in: Fall School in Analysis (Jyväskylä, 1994), T. Kilpeläinen (ed.), Report 68, Univ. Jyväskylä, Jyväskylä (1995) 33–54.Google Scholar
  13. É. Oudet: The Convex Geometry Toolbox, http://www.lama.univ-savoie.fr/~oudet/ConvGeomToolbox/ConvGeomToolbox.html.Google Scholar
  14. G. Polya, G. Szego: Isoperimetric Inequalities in Mathematical Physics, Ann. Math. Studies 27, Princeton University Press, Princeton (1951).https://doi.org/10.1515/9781400882663Google Scholar
  15. M. H. Protter: A lower bound for the fundamental frequency of a convex region, Proc. Amer. Math. Soc. 81 (1981) 65–70.https://doi.org/10.1090/S0002-9939-1981-0589137-2Google Scholar
  16. P. R. Scott, P. W. Awyong: Inequalities for convex sets, J. Inequal. Pure Appl. Math. 1(1) (2000) Article 6, 6 pp.Google Scholar
  17. Y. Yu: Some properties of the ground states of the infinity Laplacian, Indiana Univ. Math. J. 56(2) (2007) 947–964.https://doi.org/10.1512/iumj.2007.56.2935Google Scholar

Suggested citation

M. Belloni, E. Oudet. “The Minimal Gap Between Λ_2(Ω) and Λ_∞(Ω) in a Class of Convex Domains.” Journal of Convex Analysis 15 (2008), No. 3, 507–521. https://doi.org/10.68381/jca15036

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.