Publication ahead of print Issue ESAIM: COCV DOI 10.1051/cocv/2009018 Published online 02 July 2009

ESAIM: COCV
DOI: 10.1051/cocv/2009018

Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals

Pedro Freitas¹ and Batłomiej Siudeja<sup>2

1</sup>  Department of Mathematics, Faculdade de Motricidade Humana (TU Lisbon) and Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal. freitas@cii.fc.ul.pt
²  Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. siudeja@illinois.edu

Received September 18, 2008. Revised March 5, 2009. Published online July 2nd, 2009.

Abstract
We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. In particular, we improve Pólya and Szegö's [Annals of Mathematical Studies 27 (1951)] lower bound for quadrilaterals and extend Hersch's [Z. Angew. Math. Phys. 17 (1966) 457–460] upper bound for parallelograms to general quadrilaterals.

Mathematics Subject Classification. 35P15, 35J05

Key words: Dirichlet eigenvalues, polygons, variational bounds


© EDP Sciences, SMAI 2009

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