ON THE Lr-OPERATORS PENALIZED BY (r + 1)-MEAN CURVATURE.

In this paper, we establish the non-positivity of the second eigenvalue of the Schrödinger operator-div(Pr ∇ ·)-Wr² on a closed hypersurface Σn of ℝn+1, where Wr is a power of the (r + 1)-th mean curvature of Σn, 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. [ABSTRACT FROM AUTHOR]