Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Abstract

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $\varphi_\lambda: \Delta_M \varphi_\lambda + \lambda \varphi_\lambda = 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $\lambda$. We show that $$H^{n-1}(\{ \varphi_\lambda=0\}) \leq C \lambda^{\alpha},$$ where $\alpha>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index. % We obtain partial positive answers to the question: Is the frequency additive in some sense?

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  • [B] Go to document J. Brüning, "Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators," Math. Z., vol. 158, iss. 1, pp. 15-21, 1978.
    @ARTICLE{B,
      author = {Brüning, Jochen},
      title = {{Ü}ber {K}noten von {E}igenfunktionen des {L}aplace-{B}eltrami-{O}perators},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {158},
      year = {1978},
      number = {1},
      pages = {15--21},
      issn = {0025-5874},
      mrclass = {58G99 (53C20)},
      mrnumber = {0478247},
      mrreviewer = {Shukichi Tanno},
      doi = {10.1007/BF01214561},
      url = {http://dx.doi.org/10.1007/BF01214561},
      zblnumber = {0349.58012},
      }
  • [CM] Go to document T. H. Colding and W. P. Minicozzi II, "Lower bounds for nodal sets of eigenfunctions," Comm. Math. Phys., vol. 306, iss. 3, pp. 777-784, 2011.
    @ARTICLE{CM,
      author = {Colding, Tobias H. and Minicozzi, II, William P.},
      title = {Lower bounds for nodal sets of eigenfunctions},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {306},
      year = {2011},
      number = {3},
      pages = {777--784},
      issn = {0010-3616},
      mrclass = {58J50 (28A78 35P15 35P20)},
      mrnumber = {2825508},
      mrreviewer = {Julie Rowlett},
      doi = {10.1007/s00220-011-1225-x},
      url = {http://dx.doi.org/10.1007/s00220-011-1225-x},
      zblnumber = {1238.58020},
      }
  • [LM2] Go to document A. Logunov, "Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture," Ann. of Math., vol. 187, iss. 1, pp. 241-262, 2018.
    @article{LM2,
      author = {Logunov, A.},
      title = {Nodal sets of {L}aplace eigenfunctions: proof of {N}adirashvili's conjecture and of the lower bound in {Y}au's conjecture},
      journal={Ann. of Math.},
      VOLUME={187},
      number={1},
      year={2018},
      doi={10.4007/annals.2018/187.1.5},
      pages={241--262},
      }
  • [N] Go to document N. Nadirashvili, "Geometry of nodal sets and multiplicity of eigenvalues," Curr. Dev. Math., pp. 231-235, 1997.
    @ARTICLE{N,
      author = {Nadirashvili, N.},
      title = {Geometry of nodal sets and multiplicity of eigenvalues},
      journal = {Curr. Dev. Math.},
      year = {1997},
      pages = {231--235},
      doi = {10.4310/CDM.1997.v1997.n1.a16},
      }
  • [SZ] Go to document C. D. Sogge and S. Zelditch, "Lower bounds on the Hausdorff measure of nodal sets II," Math. Res. Lett., vol. 19, iss. 6, pp. 1361-1364, 2012.
    @ARTICLE{SZ,
      author = {Sogge, Christopher D. and Zelditch, Steve},
      title = {Lower bounds on the {H}ausdorff measure of nodal sets {II}},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {19},
      year = {2012},
      number = {6},
      pages = {1361--1364},
      issn = {1073-2780},
      mrclass = {58C40 (28A78 35P15 35R01)},
      mrnumber = {3091613},
      mrreviewer = {Nelia Charalambous},
      doi = {10.4310/MRL.2012.v19.n6.a14},
      url = {http://dx.doi.org/10.4310/MRL.2012.v19.n6.a14},
      zblnumber = {1283.58020},
      }
  • [Y] Go to document S. T. Yau, "Problem section," in Seminar on Differential Geometry, Princeton Univ. Press, Princeton, N.J., 1982, vol. 102, pp. 669-706.
    @INCOLLECTION{Y,
      author = {Yau, Shing Tung},
      title = {Problem section},
      booktitle = {Seminar on {D}ifferential {G}eometry},
      series = {Ann. of Math. Stud.},
      volume = {102},
      pages = {669--706},
      publisher = {Princeton Univ. Press, Princeton, N.J.},
      year = {1982},
      mrclass = {53Cxx (58-02)},
      mrnumber = {0645762},
      mrreviewer = {Yu. Burago},
      zblnumber = {0479.53001},
      doi = {10.1515/9781400881918-035},
      }
  • [AGM] S. Agmon, Unicité et convexité dans les problèmes différentiels, Les Presses de l’Université de Montréal, Montreal, Que., 1966.
    @book {AGM,
      author = {Agmon, Shmuel},
      TITLE = {Unicité et convexité dans les problèmes différentiels},
      SERIES = {Séminaire de Mathématiques Supérieures, No. 13 (Été, 1965)},
      PUBLISHER = {Les Presses de l'Université de Montréal, Montreal, Que.},
      YEAR = {1966},
      PAGES = {152},
      MRCLASS = {35.01},
      MRNUMBER = {0252808},
      MRREVIEWER = {B. Hellwig},
      ZBLNUMBER = {0147.07702},
      }
  • [ALM] F. J. Almgren Jr., "Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents," in Minimal Submanifolds and Geodesics, North-Holland, Amsterdam-New York, 1979, pp. 1-6.
    @incollection {ALM,
      author = {Almgren, Jr., Frederick J.},
      TITLE = {Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents},
      BOOKTITLE = {Minimal Submanifolds and Geodesics},
      VENUE={{P}roc. {J}apan-{U}nited {S}tates {S}em., {T}okyo, 1977},
      PAGES = {1--6},
      PUBLISHER = {North-Holland, Amsterdam-New York},
      YEAR = {1979},
      MRCLASS = {49F22},
      MRNUMBER = {574247},
      ZBLNUMBER = {0439.49028},
      }
  • [LAN] Go to document E. M. Landis, "Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables)," Uspehi Mat. Nauk, vol. 18, iss. 1 (109), pp. 3-62, 1963.
    @article {LAN,
      author = {Landis, E. M.},
      TITLE = {Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables)},
      JOURNAL = {Uspehi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      VOLUME = {18},
      YEAR = {1963},
      NUMBER = {1 (109)},
      PAGES = {3--62},
      ISSN = {0042-1316},
      MRCLASS = {35.42},
      MRNUMBER = {0150437},
      MRREVIEWER = {J. Nečas},
      ZBLNUMBER = {0122.33701},
      DOI = {10.1070/RM1963v018n01ABEH004124},
      }
  • [Cauchy] Go to document G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella, "The stability for the Cauchy problem for elliptic equations," Inverse Problems, vol. 25, iss. 12, p. 123004, 2009.
    @ARTICLE{Cauchy,
      author = {Alessandrini, Giovanni and Rondi, Luca and Rosset, Edi and Vessella, Sergio},
      title = {The stability for the {C}auchy problem for elliptic equations},
      journal = {Inverse Problems},
      fjournal = {Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data},
      volume = {25},
      year = {2009},
      number = {12},
      pages = {123004, 47},
      issn = {0266-5611},
      mrclass = {35R25 (35-02 35B35 35J05)},
      mrnumber = {2565570},
      doi = {10.1088/0266-5611/25/12/123004},
      url = {http://dx.doi.org/10.1088/0266-5611/25/12/123004},
      zblnumber = {1190.35228},
      }
  • [D] Go to document R. Dong, "Nodal sets of eigenfunctions on Riemann surfaces," J. Differential Geom., vol. 36, iss. 2, pp. 493-506, 1992.
    @ARTICLE{D,
      author = {Dong, Rui-Tao},
      title = {Nodal sets of eigenfunctions on {R}iemann surfaces},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {36},
      year = {1992},
      number = {2},
      pages = {493--506},
      issn = {0022-040X},
      mrclass = {58G25 (35P99)},
      mrnumber = {1180391},
      mrreviewer = {Stig I. Andersson},
      zblnumber = {0776.53024},
      doi = {10.4310/jdg/1214448750},
      }
  • [DF] Go to document H. Donnelly and C. Fefferman, "Nodal sets of eigenfunctions on Riemannian manifolds," Invent. Math., vol. 93, iss. 1, pp. 161-183, 1988.
    @ARTICLE{DF,
      author = {Donnelly, Harold and Fefferman, Charles},
      title = {Nodal sets of eigenfunctions on {R}iemannian manifolds},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {93},
      year = {1988},
      number = {1},
      pages = {161--183},
      issn = {0020-9910},
      mrclass = {58G25 (35B60 35P05)},
      mrnumber = {0943927},
      mrreviewer = {P. Günther},
      doi = {10.1007/BF01393691},
      url = {http://dx.doi.org/10.1007/BF01393691},
      zblnumber = {0659.58047},
      }
  • [DF1] Go to document H. Donnelly and C. Fefferman, "Nodal sets for eigenfunctions of the Laplacian on surfaces," J. Amer. Math. Soc., vol. 3, iss. 2, pp. 333-353, 1990.
    @ARTICLE{DF1,
      author = {Donnelly, Harold and Fefferman, Charles},
      title = {Nodal sets for eigenfunctions of the {L}aplacian on surfaces},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {3},
      year = {1990},
      number = {2},
      pages = {333--353},
      issn = {0894-0347},
      mrclass = {58G25 (35P05)},
      mrnumber = {1035413},
      mrreviewer = {H.-B. Rademacher},
      doi = {10.2307/1990956},
      url = {http://dx.doi.org/10.2307/1990956},
      zblnumber = {0702.58077},
      }
  • [GL] Go to document N. Garofalo and F. Lin, "Monotonicity properties of variational integrals, $A_p$ weights and unique continuation," Indiana Univ. Math. J., vol. 35, iss. 2, pp. 245-268, 1986.
    @ARTICLE{GL,
      author = {Garofalo, Nicola and Lin, Fang-Hua},
      title = {Monotonicity properties of variational integrals, {$A_p$} weights and unique continuation},
      journal = {Indiana Univ. Math. J.},
      fjournal = {Indiana University Mathematics Journal},
      volume = {35},
      year = {1986},
      number = {2},
      pages = {245--268},
      issn = {0022-2518},
      mrclass = {35J20 (35J10 42B25)},
      mrnumber = {0833393},
      mrreviewer = {Stavros A. Belbas},
      doi = {10.1512/iumj.1986.35.35015},
      url = {http://dx.doi.org/10.1512/iumj.1986.35.35015},
      zblnumber = {0678.35015},
      }
  • [HS] Go to document R. Hardt and L. Simon, "Nodal sets for solutions of elliptic equations," J. Differential Geom., vol. 30, iss. 2, pp. 505-522, 1989.
    @ARTICLE{HS,
      author = {Hardt, Robert and Simon, Leon},
      title = {Nodal sets for solutions of elliptic equations},
      journal = {J. Differential Geom.},
      fjournal = {Journal of Differential Geometry},
      volume = {30},
      year = {1989},
      number = {2},
      pages = {505--522},
      issn = {0022-040X},
      mrclass = {58E05 (35J99)},
      mrnumber = {1010169},
      mrreviewer = {Fang Hua Lin},
      zblnumber = {0692.35005},
      doi = {10.4310/jdg/1214443599},
      }
  • [HL] Q. Han and F. -H. Lin, Nodal Sets of Solutions of Elliptic Differential Equations.
    @MISC{HL,
      author = {Han, Q and Lin, F.-H.},
      title = {Nodal Sets of Solutions of Elliptic Differential Equations},
      note = {book in preparation},
      zblnumber = {},
      }
  • [L] Go to document F. Lin, "Nodal sets of solutions of elliptic and parabolic equations," Comm. Pure Appl. Math., vol. 44, iss. 3, pp. 287-308, 1991.
    @ARTICLE{L,
      author = {Lin, Fang-Hua},
      title = {Nodal sets of solutions of elliptic and parabolic equations},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {44},
      year = {1991},
      number = {3},
      pages = {287--308},
      issn = {0010-3640},
      mrclass = {58G11 (35J05 35K05 58G03)},
      mrnumber = {1090434},
      mrreviewer = {Robert McOwen},
      doi = {10.1002/cpa.3160440303},
      url = {http://dx.doi.org/10.1002/cpa.3160440303},
      zblnumber = {0734.58045},
      }
  • [LM] A. Logunov and . E. Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three.
    @MISC{LM,
      author = {Logunov, A. and Malinnikova, {\relax Eu}.},
      title = {Nodal sets of {L}aplace eigenfunctions: estimates of the {H}ausdorff measure in dimension two and three},
      note = {preprint},
      zblnumber = {},
      }
  • [M] Go to document D. Mangoubi, "The effect of curvature on convexity properties of harmonic functions and eigenfunctions," J. Lond. Math. Soc. (2), vol. 87, iss. 3, pp. 645-662, 2013.
    @ARTICLE{M,
      author = {Mangoubi, Dan},
      title = {The effect of curvature on convexity properties of harmonic functions and eigenfunctions},
      journal = {J. Lond. Math. Soc. (2)},
      fjournal = {Journal of the London Mathematical Society. Second Series},
      volume = {87},
      year = {2013},
      number = {3},
      pages = {645--662},
      issn = {0024-6107},
      mrclass = {58J50 (35J15 35P20 35R01 53C21 58E20)},
      mrnumber = {3073669},
      mrreviewer = {Tanya J. Christiansen},
      doi = {10.1112/jlms/jds067},
      url = {http://dx.doi.org/10.1112/jlms/jds067},
      zblnumber = {1316.35220},
      }
  • [NPS] Go to document F. Nazarov, L. Polterovich, and M. Sodin, "Sign and area in nodal geometry of Laplace eigenfunctions," Amer. J. Math., vol. 127, iss. 4, pp. 879-910, 2005.
    @ARTICLE{NPS,
      author = {Nazarov, Fëdor and Polterovich, Leonid and Sodin, Mikhail},
      title = {Sign and area in nodal geometry of {L}aplace eigenfunctions},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {127},
      year = {2005},
      number = {4},
      pages = {879--910},
      issn = {0002-9327},
      mrclass = {58J50 (35J25 35P20)},
      mrnumber = {2154374},
      mrreviewer = {Alessandro Savo},
      url = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v127/127.4nazarov.pdf},
      zblnumber = {1079.58026},
      doi = {10.1353/ajm.2005.0030},
      }

Authors

Alexander Logunov

School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
Chebyshev Laboratory, St. Petersburg State University, Saint Petersburg, Russia
Institute for Advanced Study, Princeton, NJ, USA