Abstract

An inequality for the reverse Bossel-Daners inequality is derived by means of the harmonic transplantation. The first and second shape derivatives are computed for the ball. The same method is then applied to elliptic boundary value problems with inhomogeneous Neumann conditions. The first variation is computed and an isoperimetric inequality is derived for the minimal energy.

Author information

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

Catherine Bandle

Mathematische Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland

Alfred Wagner

Institut für Mathematik, RWTH Aachen, Templergraben 55, 52062 Aachen, Germany

Keywords

  • Rayleigh-Faber Krahn inequality
  • Robin boundary conditions
  • domain variation
  • harmonic transplantation

Mathematics Subject Classification

49K20, 49R05, 15A42, 35J20, 35N25

References

  1. G. Auchmuty: Bases and comparison results for linear elliptic eigenproblems, J. Math. Anal. Appl. 390 (2012) 394–406.https://doi.org/10.1016/j.jmaa.2012.01.051Google Scholar
  2. G. Auchmuty: Sharp boundary trace inequalities, Proc. R. Soc. Edinb., Sect. A, Math. 144 (2014) 1–12.https://doi.org/10.1017/S0308210512000601Google Scholar
  3. C. Bandle, M. Flucher: Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations ∆U = eU and ∆U = U (n+2)/(n−2), SIAM Rev. 38 (1969) 191–238.https://doi.org/10.1137/1038039Google Scholar
  4. C. Bandle, A. Wagner: Second variaion of domain functionals and applications to problems with Robin boundary conditions, arXiv:1403.2220 [math.OC] (2014).Google Scholar
  5. M. Bareket: On an isoperimetric inequality for the first eigenvalue of a boundary value problem, SIAM J. Math. Anal. 8 (1977) 280–287.https://doi.org/10.1137/0508020Google Scholar
  6. M. H. Bossel: Membranes élastiquement liées inhomogènes ou sur une surface: une nouvelle extension du théorème isopérimétrique de Rayleigh-Faber-Krahn, Z. Angew. Math. Phys. 39 (1988) 733–742.https://doi.org/10.1007/BF00948733Google Scholar
  7. D. Daners: A Faber-Krahn inequality for Robin problems in any space dimension, Math. Ann. 333 (2006) 767–785.https://doi.org/10.1007/s00208-006-0753-8Google Scholar
  8. V. Ferone, C. Nitsch, C. Trombetti: On a conjectured reverse Faber-Krahn inequality for a Steklov-type Laplacian eigenvalue, arXiv:1307.3788 [math.OC] (2014).https://doi.org/10.3934/cpaa.2015.14.63Google Scholar
  9. P. Freitas, D. Krejcirik: The First Robin eigenvalue with a negative boundary parameter, arXiv:1403.6666 [math.SP] (2014).https://doi.org/10.1016/j.aim.2015.04.023Google Scholar
  10. J. Hersch: Transplantation harmonique, transplantation par modules, et théorèmes isopérimétriques, Comment. Math. Helv. 44 (1969) 354–366.https://doi.org/10.1007/BF02564535Google Scholar
  11. K. Pankrashkin: On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains, Nanosystems: Phys. Chem. Math. 4 (2013) 474–483.Google Scholar

Suggested citation

C. Bandle, A. Wagner. “Isoperimetric Inequalities for the Principal Eigenvalue of a Membrane and the Energy of Problems with Robin Boundary Conditions.” Journal of Convex Analysis 22 (2015), No. 3, 627–640. https://doi.org/10.68381/jca22031

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.