Stability results for the first eigenvalue of the Laplacian on domains in space forms

Abstract:

We studied the two known works on stability for isoperimetric inequalities of the first eigenvalue of the Laplacian. The earliest work is due to A. Melas who proved the stability of the Faber-Krahn inequality: for a convex domain Ω contained in ℝⁿ with λ close to ̄l, the first eigenvalue of the ball B of the same volume, the domain must be close to the ball B with respect to the Hausdorff distance. Later, Y. Xu studied the stability of the Szegö-Weinberger inequality for convex domains in ℝⁿ and ℍⁿ where ℍⁿ denotes hyperbolic space. Our work consists of extending A. Melas' result to the spaces of constant curvature S² and ℍ² and Y. Xu's result to domains contained in the polar cap B<inf>π/4</inf> in Sⁿ. © 2002 Elsevier Science (USA).

Año de publicación:

2002

Keywords:

• Stability of eigenvalues
• Szegö-Weinberger inequality
• Faber-Krahn inequality
• Space forms
• Constant curvature

Fuente:

scopus