×

zbMATH — the first resource for mathematics

Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture. (English) Zbl 1384.58021
Let \(\mathcal{M}=(M,g)\) be a compact closed smooth Riemannian manifold of dimension \(n\). Let \(\Delta\) be the scalar Laplacian and let \(\phi_\lambda\) be a non-trivial eigenfunction of the Laplacian. The author shows that there exists \(c(\mathcal{M})>0\) so that \(c\lambda^{\frac12}\leq H^{n-1}\{x:\phi_\lambda(x)=0\}\) where \(H^{n-1}\) denotes \((n-1)\) dimensional Hausdorf measure. An analogous upper bound is established in a previous paper by the author. The author also establishes a similar estimate for harmonic functions in the unit ball of \(\mathbb{R}^n\) which establishes a conjecture by N. Nadirashvili [A. Jaffe et al., in: Current developments in mathematics, 1997. Papers from the conference held in Cambridge, MA, USA, 1997. Boston, MA: International Press. 189–266 (1999; Zbl 1031.00501)].
The introduction states the main results and provides historical contextualization. The second section treats almost monotonicity of the frequency. The third section gives a lemma concerning monotonic functions. The fourth section discusses the behaviour near the maximum. The fifth section deals with the number of cubes with big doubling index. The sixth section contains a geometrical construction which is useful for providing lower estimates for the nodal sets. The seventh section estimates the volume of the nodal set and establishes the desired estimate on the ball. The eighth section provides a lower bound in Yau’s conjecture and deals with a general Riemannian manifold.

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P05 General topics in linear spectral theory for PDEs
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Yau, Shing Tung, Problem section. Seminar on {D}ifferential {G}eometry, Ann. of Math. Stud., 102, 669-706, (1982) · Zbl 0479.53001
[2] Donnelly, Harold; Fefferman, Charles, Nodal sets of eigenfunctions on {R}iemannian manifolds, Invent. Math.. Inventiones Mathematicae, 93, 161-183, (1988) · Zbl 0659.58047
[3] Donnelly, Harold; Fefferman, Charles, Nodal sets for eigenfunctions of the {L}aplacian on surfaces, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 3, 333-353, (1990) · Zbl 0702.58077
[4] Garofalo, Nicola; Lin, Fang-Hua, Monotonicity properties of variational integrals, {\(A_p\)} weights and unique continuation, Indiana Univ. Math. J.. Indiana University Mathematics Journal, 35, 245-268, (1986) · Zbl 0678.35015
[5] Hardt, Robert; Simon, Leon, Nodal sets for solutions of elliptic equations, J. Differential Geom.. Journal of Differential Geometry, 30, 505-522, (1989) · Zbl 0692.35005
[6] Han, Q.; Lin, F.-H., Nodal Sets of Solutions of Elliptic Differential Equations
[7] Logunov, A.; Malinnikova, {\relax Eu}., Nodal sets of {L}aplace eigenfunctions: estimates of the {H}ausdorff measure in dimension two and three
[8] Logunov, A., Nodal sets of {L}aplace eigenfunctions: polynomial upper bounds for the {H}ausdorff measure, Ann. of Math., 187, 221-239, (2018) · Zbl 1384.58020
[9] Colding, Tobias H.; Minicozzi, II, William P., Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys.. Communications in Mathematical Physics, 306, 777-784, (2011) · Zbl 1238.58020
[10] Sogge, Christopher D.; Zelditch, Steve, Lower bounds on the {H}ausdorff measure of nodal sets {II}, Math. Res. Lett.. Mathematical Research Letters, 19, 1361-1364, (2012) · Zbl 1283.58020
[11] Br\`“uning, Jochen, {\'”{U}}ber {K}noten von {E}igenfunktionen des {L}aplace-{B}eltrami-{O}perators, Math. Z.. Mathematische Zeitschrift, 158, (1978) · Zbl 0349.58012
[12] Gilbarg, David; Trudinger, Neil S., Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., 224, x+401 pp., (1977) · Zbl 0361.35003
[13] Lin, Fang-Hua, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 44, 287-308, (1991) · Zbl 0734.58045
[14] Mangoubi, Dan, The effect of curvature on convexity properties of harmonic functions and eigenfunctions, J. Lond. Math. Soc. (2). Journal of the London Mathematical Society. Second Series, 87, 645-662, (2013) · Zbl 1316.35220
[15] Nadirashvili, N., Geometry of nodal sets and multiplicity of eigenvalues, Curr. Dev. Math., 231-235, (1997) · Zbl 1384.58022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.