Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T08:14:46.706Z Has data issue: false hasContentIssue false
  • English
  • Français

A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators∗

Published online by Cambridge University Press:  11 May 2012

Qi Lü
Qi Lü*
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, P.R. China. luqi59@163.com Basque Center for Applied Mathematics (BCAM), Mazarredo, 14, 48009 Bilbao Basque Country, Spain
Get access

Abstract

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.

Keywords

Lower boundlocal energypartial sum of eigenfunctionsLaplace-Beltrami operatorRobin boundary condition
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 255 - 273
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Fu, X., A weighted identity for partial differential operators of second order and its applications. C. R. Acad. Sci., Sér. I Paris 342 (2006) 579–584. Google Scholar
A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser. Seoul National University, Seoul 34 (1996).
E. Hebey, Nonlinear Analysis on Manifolds : Sobolev Spaces and Inequalities, Courant Lect. Notes Math. New York University Courant Institute of Mathematical Sciences, New York 5 (1999).
D. Jerison and G. Lebeau, Nodal sets of sums of eigenfunctions, in Harmonic Analysis and Partial Differential Equations. Chicago, IL (1996) 223–239; Chicago Lect. Math., Univ. Chicago Press, Chicago, IL (1999).
J. Jost, Riemann Geometry and Geometric Analysis. Springer-Verlag, Berlin, Heidelberg (2005).
M.M. Larent’ev, V.G. Romanov and S.P. Shishat·Skii, Ill-posed Problems of Mathematical Physics and Analysis. Edited by Amer. Math. Soc. Providence. Transl. Math. Monogr. 64 (1986).
Lebeau, G. and Robbiano, L., Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335–356. Google Scholar
Lebeau, G. and Robbiano, L., Stabilizzation de l’équation des ondes par le bord. Duke Math. J. 86 (1997) 465–491. Google Scholar
Lebeau, G. and Zuazua, E., Null controllability of a system of linear thermoelasticity. Arch. Ration. Mech. Anal. 141 (1998) 297–329. Google Scholar
Liu, X. and Zhang, X., On the local controllability of a class of multidimensional quasilinear parabolic equations. C. R. Math. Acad. Sci., Paris 347 (2009) 1379–1384. Google Scholar
López, A., Zhang, X. and Zuazua, E., Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations. J. Math. Pure. Appl. 79 (2000) 741–808. Google Scholar
Lü, Q., Bang-Bang principle of time optimal controls and null controllability of fractional order parabolic equations. Acta Math. Sin. 26 (2010) 2377–2386. Google Scholar
Q. Lü, Control and Observation of Stochastic Partial Differential Equations. Ph.D. thesis, Sichuan University (2010).
Lü, Q., Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct. Anal. 260 (2011) 832–851. Google Scholar
Lü, Q. and Wang, G., On the existence of time optimal controls with constraints of the rectangular type for heat equations. SIAM J. Control Optim. 49 (2011) 1124–1149. Google Scholar
Miller, L., How violent are fast controls for Schrödinger and plate vibrations? Arch. Ration. Mech. Anal. 172 (2004) 429–456. Google Scholar
J. Milnor, Morse Theory, Ann. Math. Studies. Princeton Univ. Press, Princeton, NJ (1963).
Phung, K.-D. and Zhang, X., Time reversal focusing of the initial state for Kirchhoff plate. SIAM J. Appl. Math. 68 (2008) 1535–1556. Google Scholar
Wang, G., L ∞-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim. 47 (2008) 1701–1720. Google Scholar
Zhang, X., Explicit observability estimate for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim. 39 (2001) 812–834. Google Scholar
Zheng, C., Controllability of the time discrete heat equation. Asymptot. Anal. 59 (2008) 139–177. Google Scholar
Zuazua, E., Controllability and observability of partial differential equations : Some results and open problems, in Handbook of Differential Equations : Evolutionary Differential Equations 3 (2006) 527–621. Google Scholar