Abstract
Let eλ(x) be a Neumann eigenfunction with respect to the positive Laplacian Δ on a compact Riemannian manifold M with boundary such that Δe λ = λ2 e λ in the interior of M and the normal derivative of e λ vanishes on the boundary of M. Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M. The authors show the following gradient estimate for χλ f as \(\lambda \geqslant 1:{\left\| {\nabla {\chi _\lambda }{\kern 1pt} \left. f \right\|} \right._\infty } \leqslant C\left( {\lambda \left\| {{\chi _\lambda }{{\left. f \right\|}_\infty } + \left. {{\lambda ^{ - 1}}} \right\|} \right.\Delta {\chi _\lambda }{{\left. f \right\|}_\infty }} \right)\), where C is a positive constant depending only on M. As a corollary, the authors obtain the gradient estimate of eλ: For every λ ≥ 1, it holds that \(\left\| {\nabla {{\left. {{e_\lambda }} \right\|}_\infty } \leqslant C\left. \lambda \right\|} \right.{\left. {{e_\lambda }} \right\|_\infty }\).
Similar content being viewed by others
References
Brandt, A., Interior estimates for second order elliptic differential (or finite difference) equations via the maximum principle, Israel J. Math., 7, 1969, 95–121.
Brüning, J., Über knoten von eigenfunktionen des Laplace-Beltrami operators, Math. Z., 158, 1975, 15–21.
Donnelly, H. and Fefferman, C., Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93(1), 1988, 161–183.
Donnelly, H. and Fefferman, C., Growth and geometry of eigenfunctions of the Laplacian, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., New York, 122, 1990, 635–655.
Duong, X. T., Ouhabaz, E. M. and Sikora, A., Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196, 2002, 443–485.
Gilbarg, D. and Trudinger, Neil S., Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 Edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
Grieser, D., Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Comm. Partial Differential Equations, 27(7–8), 2002, 1283–1299.
Hörmander, L., The Analysis of Linear Partial Differential Equations III, Corrected Second Printing, Springer-Verlag, Tokyo, 1994.
Seeger, A. and Sogge, C. D., On the boundedness of functions of pseudo-differential operators on a compact manifolds, Duke Math. J., 59, 1989, 709–736.
Shi, Y. Q. and Xu, B., Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary, Ann. Glob. Anal. Geom., 2010, 38(1), 21–26.
Shi, Y. Q. and Xu, B., Gradient estimate of a Dirichlet eigenfunction on a compact manifold with boundary, Forum Math., DOI: 10.1515/FORM.2011.115
Smith, H. F., Sharp L 2 → L q bounds on the spectral projectors for low regularity metrics, Math. Res. Lett., 13(5–6), 2006, 967–974.
Xu, X., Gradient estimates for eigenfunctions of compact manifolds with boundary and the Hörmander multiplier theorem, Forum Mathematicum, 21(3), 2009, 455–476.
Xu, X., Eigenfunction estimates for Neumann Laplacian and applications to multiplier problems, Proc. Amer. Math. Soc., 139, 2011, 3583–3599.
Zelditch, S., Local and global analysis of eigenfunctions on Riemannian manifolds, Handbook of geometric analysis, No. 1, 545–658, Adv. Lect. Math., 7, Int. Press, Somerville, MA, 2008.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 10971104, 11271343, 11101387), the Anhui Provincial Natural Science Foundation (No. 1208085MA01) and the Fundamental Research Funds for the Central Universities (Nos.WK0010000020, WK0010000023, WK3470000003).
Rights and permissions
About this article
Cite this article
Hu, J., Shi, Y. & Xu, B. The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary. Chin. Ann. Math. Ser. B 36, 991–1000 (2015). https://doi.org/10.1007/s11401-015-0924-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-015-0924-6