Differential operators penalized by geometric potentials.

Abstract

This paper is presented in two parts. In the first part, we establish the non-positivity of the second eigenvalue of the Schrödinger operator −div P r ∇ · − W 2r on a closed hypersurface Σ n of Rn+1 , where W r is a power of the (r + 1)-th mean curvature of Σ n which we will ask to be positive. If this eigenvalue is null, we will have a characterization of the sphere. This theorem generalizes the result of Harrell and Loss proved to the Laplace-Beltrame operator penalized by the square of the mean curvature. In the second part, we established the non-positivity of the second auto-value of the Schödinger operator − d2ds2 − (√F) −2CF(κ), in a closed curve of the plane with length 2π, F ∈ C 1 ( R ) and κ is the curvature of the curve. If this eigenvalue is null, we will have a characterization of the circle, which generalizes partially the result of Harrell and Loss proved to the one-dimensional Laplace operator penalized by the square of the curvature in curves of the plane.