New Bounds for the Principal Dirichlet Eigenvalue of Planar Regions

Abstract

We present a numerical study for the first Dirichlet eigenvalue of certain classes of planar regions. Based on this, we propose new types of bounds and establish several conjectures regarding the dependence of this eigenvalue on the perimeter and the area.

Keywords:

• Primary 35P15
• Secondary 58G25
• Dirichlet eigenvalues
• Pólya–Szeg½o conjectures