A Faber-Krahn Inequality for Solutions of Schrodinger's Equation on Riemannian Manifolds

作者：Abreu Emerson; Barbosa Ezequiel^*

来源：Journal of Geometric Analysis, 2018, 28(2): 1078-1090.

DOI：10.1007/s12220-017-9854-8

• Faber-Krahn
• Schrodinger
• Bounded potential
• First eigenvalue
• Isoperimetric inequality

We consider a bounded open set with smooth boundary Omega subset of M in a Riemannian manifold (M, g), and suppose that there exists a non-trivial function u is an element of C((Omega) over bar) solving the problem
-Delta u = V(x) u, in Omega,
in the distributional sense, with V is an element of L-infinity(Omega), where u = 0 on partial derivative Omega. We prove a sharp inequality involving parallel to V parallel to(L infinity)(Omega) and the first eigenvalue of the Laplacian on geodesic balls in simply connected spaces with constant curvature, which slightly generalises the well-known Faber-Krahn isoperimetric inequality. Moreover, in a Riemannian manifold which is not necessarily simply connected, we obtain a lower bound for parallel to V parallel to(L infinity)(Omega) in terms of its isoperimetric or Cheeger constant. As an application, we show that if Omega is a domain on a m-dimensional minimal submanifold of R-n which lies in a ball of radius R, then
parallel to V parallel to(L infinity)(Omega) >= (m/2R)(2).

• 出版日期2018-4