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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/cocv/2012008/

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