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 bound, local energy, partial sum of eigenfunctions, Laplace-Beltrami operator, Robin boundary condition

@article{COCV_2013__19_1_255_0, author = {L\"u, Qi}, title = {A lower bound on local energy of partial sum of eigenfunctions for {Laplace-Beltrami} operators}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {255--273}, publisher = {EDP-Sciences}, volume = {19}, number = {1}, year = {2013}, doi = {10.1051/cocv/2012008}, mrnumber = {3023069}, zbl = {1258.93035}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012008/} }

TY - JOUR AU - Lü, Qi TI - A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 255 EP - 273 VL - 19 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012008/ DO - 10.1051/cocv/2012008 LA - en ID - COCV_2013__19_1_255_0 ER -

%0 Journal Article %A Lü, Qi %T A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 255-273 %V 19 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012008/ %R 10.1051/cocv/2012008 %G en %F COCV_2013__19_1_255_0

Lü, Qi. A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 1, pp. 255-273. doi : 10.1051/cocv/2012008. http://www.numdam.org/articles/10.1051/cocv/2012008/

[1] A weighted identity for partial differential operators of second order and its applications. C. R. Acad. Sci., Sér. I Paris 342 (2006) 579-584. | MR

,[2] Controllability of Evolution Equations, Lect. Notes Ser. Seoul National University, Seoul 34 (1996). | MR | Zbl

and ,[3] Nonlinear Analysis on Manifolds : Sobolev Spaces and Inequalities, Courant Lect. Notes Math. New York University Courant Institute of Mathematical Sciences, New York 5 (1999). | MR | Zbl

,[4] 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). | MR | Zbl

and ,[5] Riemann Geometry and Geometric Analysis. Springer-Verlag, Berlin, Heidelberg (2005). | MR | Zbl

,[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). | MR | Zbl

[7] Contrôle exact de l'équation de la chaleur. Commun. Partial Differ. Equ. 20 (1995) 335-356. | MR | Zbl

and ,[8] Stabilizzation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465-491. | MR | Zbl

and ,[9] Null controllability of a system of linear thermoelasticity. Arch. Ration. Mech. Anal. 141 (1998) 297-329. | MR | Zbl

and ,[10] On the local controllability of a class of multidimensional quasilinear parabolic equations. C. R. Math. Acad. Sci., Paris 347 (2009) 1379-1384. | MR | Zbl

and ,[11] 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. | MR | Zbl

, and ,[12] Bang-Bang principle of time optimal controls and null controllability of fractional order parabolic equations. Acta Math. Sin. 26 (2010) 2377-2386. | MR | Zbl

,[13] Control and Observation of Stochastic Partial Differential Equations. Ph.D. thesis, Sichuan University (2010).

,[14] Some results on the controllability of forward stochastic heat equations with control on the drift. J. Funct. Anal. 260 (2011) 832-851. | MR | Zbl

,[15] On the existence of time optimal controls with constraints of the rectangular type for heat equations. SIAM J. Control Optim. 49 (2011) 1124-1149. | MR | Zbl

and ,[16] How violent are fast controls for Schrödinger and plate vibrations? Arch. Ration. Mech. Anal. 172 (2004) 429-456. | MR | Zbl

,[17] Morse Theory, Ann. Math. Studies. Princeton Univ. Press, Princeton, NJ (1963). | MR | Zbl

,[18] Time reversal focusing of the initial state for Kirchhoff plate. SIAM J. Appl. Math. 68 (2008) 1535-1556. | MR | Zbl

and ,[19] L∞-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim. 47 (2008) 1701-1720. | MR | Zbl

,[20] Explicit observability estimate for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim. 39 (2001) 812-834. | MR | Zbl

,[21] Controllability of the time discrete heat equation. Asymptot. Anal. 59 (2008) 139-177. | MR | Zbl

,[22] Controllability and observability of partial differential equations : Some results and open problems, in Handbook of Differential Equations : Evolutionary Differential Equations 3 (2006) 527-621. | MR | Zbl

,*Cited by Sources: *