Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds

Content Provider	Semantic Scholar
Author	Fall, Mouhamed Moustapha Weth, Tobias
Copyright Year	2011
Abstract	We study the local Szegö–Weinberger profile in a geodesic ball $$B_g(y_0,r_0)$$Bg(y0,r0) centered at a point $$y_0$$y0 in a Riemannian manifold $$({\mathcal {M}},g)$$(M,g). This profile is obtained by maximizing the first nontrivial Neumann eigenvalue $$\mu _2$$μ2 of the Laplace–Beltrami Operator $$\Delta _g$$Δg on $${\mathcal {M}}$$M among subdomains of $$B_g(y_0,r_0)$$Bg(y0,r0) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of $${\mathcal {M}}$$M at $$y_0$$y0. As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of $$\Delta _g$$Δg, but additional difficulties arise due to the fact that $$\mu _2$$μ2 is degenerate in the unit ball in $$\mathbb {R}^N$$RN and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.
Starting Page	217
Ending Page	242
Page Count	26
File Format	PDF HTM / HTML
DOI	10.1007/s00526-013-0672-y
Alternate Webpage(s)	https://arxiv.org/pdf/1110.4770v3.pdf
Alternate Webpage(s)	https://doi.org/10.1007/s00526-013-0672-y
Volume Number	51
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article