Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals

Issue		ESAIM: COCV Volume 16, Number 3, July-September 2010


Page(s)		648 - 676
DOI		10.1051/cocv/2009018
Published online		02 July 2009

ESAIM: COCV 16 (2010) 648-676

Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals

Pedro Freitas¹ and Batłomiej Siudeja²

¹ 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: 18 September 2008
Revised: 5 March 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.

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