Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Lower bounds for interior nodal sets of Steklov eigenfunctions


Authors: Christopher D. Sogge, Xing Wang and Jiuyi Zhu
Journal: Proc. Amer. Math. Soc. 144 (2016), 4715-4722
MSC (2010): Primary 35-xx
DOI: https://doi.org/10.1090/proc/13067
Published electronically: July 22, 2016
MathSciNet review: 3544523
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the interior nodal sets, $ Z_\lambda $ of Steklov eigenfunctions in an $ n$-dimensional relatively compact manifold $ M$ with boundary and show that one has the lower bounds $ \vert Z_\lambda \vert\ge c\lambda ^{\frac {2-n}2}$ for the size of its $ (n-1)$-dimensional Hausdorff measure. The proof is based on a Dong-type identity and estimates for the gradient of Steklov eigenfunctions, similar to those in previous works of the first author and Zelditch.


References [Enhancements On Off] (What's this?)

  • [1] Katarína Bellová and Fang-Hua Lin, Nodal sets of Steklov eigenfunctions, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 2239-2268. MR 3396451, https://doi.org/10.1007/s00526-015-0864-8
  • [2] Jochen Brüning, Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators, Math. Z. 158 (1978), no. 1, 15-21 (German). MR 0478247
  • [3] Alberto-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65-73. MR 590275
  • [4] Tobias H. Colding and William P. Minicozzi II, Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys. 306 (2011), no. 3, 777-784. MR 2825508, https://doi.org/10.1007/s00220-011-1225-x
  • [5] Rui-Tao Dong, Nodal sets of eigenfunctions on Riemann surfaces, J. Differential Geom. 36 (1992), no. 2, 493-506. MR 1180391
  • [6] Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161-183. MR 943927, https://doi.org/10.1007/BF01393691
  • [7] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
  • [8] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • [9] A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, arXiv:1411.6567.
  • [10] Q. Han and F.H. Lin, Nodal sets of solutions of elliptic differential equations, book in preparation (online at http://www.nd.edu/qhan/nodal.pdf).
  • [11] Hamid Hezari and Christopher D. Sogge, A natural lower bound for the size of nodal sets, Anal. PDE 5 (2012), no. 5, 1133-1137. MR 3022851, https://doi.org/10.2140/apde.2012.5.1133
  • [12] Alessandro Savo, Lower bounds for the nodal length of eigenfunctions of the Laplacian, Ann. Global Anal. Geom. 19 (2001), no. 2, 133-151. MR 1826398, https://doi.org/10.1023/A:1010774905973
  • [13] A. Seeger and C. D. Sogge, Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J. 38 (1989), no. 3, 669-682. MR 1017329, https://doi.org/10.1512/iumj.1989.38.38031
  • [14] Yiqian Shi and Bin Xu, Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary, Ann. Global Anal. Geom. 38 (2010), no. 1, 21-26. MR 2657840, https://doi.org/10.1007/s10455-010-9198-0
  • [15] Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579
  • [16] Christopher D. Sogge, Concerning the $ L^p$ norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123-138. MR 930395, https://doi.org/10.1016/0022-1236(88)90081-X
  • [17] W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique, Ann. Sci. École Norm. Sup. (3) 19 (1902), 191-259 (French). MR 1509012
  • [18] Christopher D. Sogge and Steve Zelditch, Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett. 18 (2011), no. 1, 25-37. MR 2770580, https://doi.org/10.4310/MRL.2011.v18.n1.a3
  • [19] Christopher D. Sogge and Steve Zelditch, Lower bounds on the Hausdorff measure of nodal sets II, Math. Res. Lett. 19 (2012), no. 6, 1361-1364. MR 3091613, https://doi.org/10.4310/MRL.2012.v19.n6.a14
  • [20] Xing Wang and Jiuyi Zhu, A lower bound for the nodal sets of Steklov eigenfunctions, Math. Res. Lett. 22 (2015), no. 4, 1243-1253. MR 3391885
  • [21] Xiangjin Xu, Gradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander multiplier theorem, Forum Math. 21 (2009), no. 3, 455-476. MR 2526794, https://doi.org/10.1515/FORUM.2009.021
  • [22] Shing Tung Yau, Survey on partial differential equations in differential geometry, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 3-71. MR 645729
  • [23] S. Zelditch, Measure of nodal sets of analytic Steklov eigenfunctions, arXiv:1403.0647.
  • [24] Jiuyi Zhu, Doubling property and vanishing order of Steklov eigenfunctions, Comm. Partial Differential Equations 40 (2015), no. 8, 1498-1520. MR 3355501, https://doi.org/10.1080/03605302.2015.1025980

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35-xx

Retrieve articles in all journals with MSC (2010): 35-xx


Additional Information

Christopher D. Sogge
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email: sogge@jhu.edu

Xing Wang
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Address at time of publication: Department of Mathematics, Wayne State University, Detroit, MI 48202
Email: fz1316@wayne.edu

Jiuyi Zhu
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Address at time of publication: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803
Email: zhu@math.isu.edu

DOI: https://doi.org/10.1090/proc/13067
Received by editor(s): March 16, 2015
Published electronically: July 22, 2016
Additional Notes: The first two authors were supported in part by the NSF grant DMS-1361476
The third author was supported in part by the NSF grant DMS-1500468
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society