New Bounds for the Principal Dirichlet Eigenvalue of Planar Regions

Abstract

CONTENTS 1. Introduction 2. The Numerical Method 3. Results for λ 1 on Polygons 4. One-Term Bounds 5. Two-Term Bounds 6. Discussion Acknowledgments References

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

“…For 2p 2 ≤ l 1 ≤ 4p 2 √ 3/3 the upper boundary is defined by a curve of quadrilaterals with one symmetry (having l 2 = l 3 ), which connects the square to the equilateral triangle. The fact that the rectangles and some isosceles triangles appear as extremal sets has already been detected by Antunes & Freitas (2006) when studying isoperimetric bounds for l 1 (U) and by Antunes & Freitas (2008) for the spectral gap, l 2 (U) − l 1 (U).…”

Maximizing Neumann fundamental tones of triangles

“…Freitas [13] showed for arbitrary triangles that λ 1 A 2 S 2 ≤ π 2 3 , which is slightly weaker than (10.1); quadrilaterals have been studied too [16]. Conjectures involving λ 1 and geometric functionals have been raised by Antunes and Freitas [1].…”

“…As in the earlier proofs, we reduce to considering the triangle T with vertices (−1, 0), (1,0) Consider the polynomial trial functions…”

“…These last bounds can be strengthened to include the isoperimetric excess; see Siudeja [34,Conjecture 1.2] and Freitas and Antunes [1].…”

“…Dirichlet eigenvalues of triangles have received considerable attention [1,2,13,16,24,34,35], as discussed in Section 10. Dirichlet eigenvalues of degenerate domains have also been investigated lately [8,14].…”