Sharp estimates for Dirichlet eigenfunctions in horn-shaped regions (Q1207994)

The purpose of the paper is to prove sharp exponential bounds for the decay of the Dirichlet eigenfunctions in horn-shaped regions in the plane. The estimate is obtained using a method of Carleman [\textit{Sh. Agmon}, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of \(N\)-body Schrödinger operators (1982; Zbl 0503.35001)] and can be used to prove intrinsic ultracontractivity for the Dirichlet Laplacian for these regions. The result sharpens Theorem 7.3 in the article of \textit{V. V. Kornienko} [Differ. Uravn. 21, No. 1, 65-77 (1985; Zbl 0573.35068)].