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

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1. [1] X. Fu, A weighted identity for partial differential operators of second order and its applications. C. R. Acad. Sci., Sér. I Paris 342 (2006) 579–584. Zbl05045951MR2217919
2. [2] A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, Lect. Notes Ser. Seoul National University, Seoul 34 (1996). Zbl0862.49004MR1406566
3. [3] 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). Zbl0981.58006MR1688256
4. [4] 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). Zbl0946.35055MR1743865
5. [5] J. Jost, Riemann Geometry and Geometric Analysis. Springer-Verlag, Berlin, Heidelberg (2005). Zbl0828.53002MR2165400
6. [6] 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). Zbl0593.35003MR847715
7. [7] G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ.20 (1995) 335–356. Zbl0819.35071MR1312710
8. [8] G. Lebeau and L. Robbiano, Stabilizzation de l’équation des ondes par le bord. Duke Math. J.86 (1997) 465–491. Zbl0884.58093MR1432305
9. [9] G. Lebeau and E. Zuazua, Null controllability of a system of linear thermoelasticity. Arch. Ration. Mech. Anal.141 (1998) 297–329. Zbl1064.93501MR1620510
10. [10] X. Liu and X. Zhang, On the local controllability of a class of multidimensional quasilinear parabolic equations. C. R. Math. Acad. Sci., Paris 347 (2009) 1379–1384. Zbl1180.35296MR2588785
11. [11] A. López, X. Zhang and E. Zuazua, 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. Zbl1079.35017MR1782102
12. [12] Q. Lü, Bang-Bang principle of time optimal controls and null controllability of fractional order parabolic equations. Acta Math. Sin.26 (2010) 2377–2386. Zbl1209.49008MR2737308
13. [13] Q. Lü, Control and Observation of Stochastic Partial Differential Equations. Ph.D. thesis, Sichuan University (2010). 
14. [14] Q. Lü, Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct. Anal.260 (2011) 832–851. Zbl1213.60105MR2737398
15. [15] Q. Lü and G. Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations. SIAM J. Control Optim.49 (2011) 1124–1149. Zbl1228.49005MR2806578
16. [16] L. Miller, How violent are fast controls for Schrödinger and plate vibrations? Arch. Ration. Mech. Anal.172 (2004) 429–456. Zbl1067.35130MR2062431
17. [17] J. Milnor, Morse Theory, Ann. Math. Studies. Princeton Univ. Press, Princeton, NJ (1963). Zbl0108.10401MR163331
18. [18] K.-D. Phung and X. Zhang, Time reversal focusing of the initial state for Kirchhoff plate. SIAM J. Appl. Math.68 (2008) 1535–1556. Zbl1156.35103MR2424951
19. [19] G. Wang, L∞-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim.47 (2008) 1701–1720. Zbl1165.93016MR2421326
20. [20] X. Zhang, Explicit observability estimate for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim.39 (2001) 812–834. Zbl0982.35059MR1786331
21. [21] C. Zheng, Controllability of the time discrete heat equation. Asymptot. Anal.59 (2008) 139–177. Zbl1154.93007MR2450357
22. [22] E. Zuazua, Controllability and observability of partial differential equations : Some results and open problems, in Handbook of Differential Equations : Evolutionary Differential Equations 3 (2006) 527–621. Zbl1193.35234MR2549374