Abstract
Let \(\Omega \subset \mathbf{R}^{n}\) be a bounded domain with boundary of class \(\mathcal{C}^{1}\). One can measure various geometric and physical quantities attached to \(\Omega\), such as volume, perimeter, diameter, in-radius, torsional rigidity, and principal frequency. The first chapter of [16] contains a long list of such interesting quantities, as well as their values for standard shapes such as disks, rectangles, strips, and triangles.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Brasco, On torsional rigidity and principal frequencies: an invitation to the Kohler–Jobin rearrangement technique. ESAIM Control Optim. Calc. Var. (2013). https://hal.archives-ouvertes.fr/hal-00783875
R. Burckel, D. Marshall, D. Minda, P. Poggi-Corradini, T. Ransford, Area, capacity, and diameter versions of Schwarz’s lemma. Conform. Geom. Dyn. 12, 133–151 (2008)
T. Carroll, J. Ratzkin, Interpolating between torsional rigidity and principal frequency. J. Math. Anal. Appl. 379, 818–826 (2011)
T. Carroll, J. Ratzkin, Two isoperimetric inequalities for the Sobolev constant. Z. Angew. Math. Phys. 63, 855–863 (2012)
T. Carroll, J. Ratzkin, An isoperimetric inequality for extremal Sobolev functions. RIMS Kôkyûroku Bessatsu (to appear)
G. Chiti, A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators. Z. Angew. Math. Phys. 33, 143–148 (1982)
A. Colesanti, Brunn–Minkowski inequalities for variational functionals and related problems. Adv. Math. 194, 105–140 (2005)
Q. Dai, R. He, H. Hu, Isoperimetric inequalities and sharp estimates for positive solution of sublinear elliptic equations (Preprint, 2010). arXiv:AP/1003.3768
S.E. Graversen, M. Rao, Brownian motion and eigenvalues for the Dirichlet Laplacian. Math. Z. 203, 699–708 (1990)
J. Hersch, Sur la fréquence fondamentale d’une membrane vibrante: évaluations par défault et principe de maximum. Z. Angew. Math. Mech. 11, 387–413 (1960)
M.T. Kohler-Jobin, Sur la première fonction propre d’une membrane: une extension à N dimensions de l’inégalité isopérimétrique de Payne-Rayner. Z. Angew. Math. Phys. 28, 1137–1140 (1977)
R. Laugesen, C. Morpurgo, Extremals for eigenvalues of Laplacians under conformal mappings. J. Funct. Anal. 155, 64–108 (1998)
P. Lindqvist, On non-linear Rayleigh quotients. Potential Anal. 2, 199–218 (1993)
L.E. Payne, M.E. Rayner, An isoperimetric inequality for the first eigenfunction in the fixed membrane problem. Z. Angew. Math. Phys. 23, 13–15 (1972)
L.E. Payne, M.E. Rayner, Some isoperimetric norm bounds for solutions of the Helmholtz equation. Z. Angew. Math. Phys. 24, 105–110 (1973)
G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics (Princeton University Press, Princeton, 1951)
R. Sperb, Maximum Principles and Their Applications (Academic, New York, 1981)
P. Topping, The isoperimetric inequality on a surface. Manuscr. Math. 100, 23–33 (1999)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Ratzkin, J., Carroll, T. (2015). Isoperimetric Inequalities for Extremal Sobolev Functions. In: González, M., Yang, P., Gambino, N., Kock, J. (eds) Extended Abstracts Fall 2013. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-21284-5_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-21284-5_13
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-21283-8
Online ISBN: 978-3-319-21284-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)