Let γ be a smooth curve whose image is symmetric with respect to the y-axis, and let D be a planar domain consisting of the points on one side of γ, within a suitable distance δ of γ. Denote by μ^odd(D) the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the y-axis. If γ satisfies some simple geometric conditions, then μ^odd(D) can be sharply estimated from below in terms of the length of γ, its curvature, and δ. Moreover, we give explicit conditions on δ that ensure μ^odd(D) = μ (D).
Optimal lower bounds for eigenvalues of Neumann problems in non-convex domains / Brandolini, Barbara. - (2016). (Intervento presentato al convegno 9th European Conference on Elliptic and Parabolic Problems tenutosi a Gaeta nel 26 maggio 2016).
Optimal lower bounds for eigenvalues of Neumann problems in non-convex domains
BRANDOLINI, BARBARA
2016
Abstract
Let γ be a smooth curve whose image is symmetric with respect to the y-axis, and let D be a planar domain consisting of the points on one side of γ, within a suitable distance δ of γ. Denote by μ^odd(D) the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the y-axis. If γ satisfies some simple geometric conditions, then μ^odd(D) can be sharply estimated from below in terms of the length of γ, its curvature, and δ. Moreover, we give explicit conditions on δ that ensure μ^odd(D) = μ (D).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
