A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators∗
1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, P.R. China
2 Basque Center for Applied Mathematics (BCAM), Mazarredo, 14, 48009 Bilbao Basque Country, Spain
Received: 28 October 2011
Revised: 2 February 2012
In this paper, a lower bound is established for the local energy of partial sum of eigenfunctions for Laplace-Beltrami operators (in Riemannian manifolds with low regularity data) with general boundary condition. This result is a consequence of a new pointwise and weighted estimate for Laplace-Beltrami operators, a construction of some nonnegative function with arbitrary given critical point location in the manifold, and also two interpolation results for solutions of elliptic equations with lateral Robin boundary conditions.
Mathematics Subject Classification: 93B07
Key words: Lower bound / local energy / partial sum of eigenfunctions / Laplace-Beltrami operator / Robin boundary condition
This work is partially supported by the NSF of China under Grants 10831007, 11101070 and 60974035. This paper is an improved version of one chapter of the author’s Ph.D. thesis  accomplished at Sichuan University under the guidance of Professor Xu Zhang. The author would like to take this opportunity to thank him deeply for his help.
© EDP Sciences, SMAI, 2012