Abstract
We show that a second Neumann eigenfunction $u$ of a Euclidean triangle has at most one (non-vertex) critical point $p$, and if $p$ exists, then it is a non-degenerate critical point of Morse index $1$. Using this we deduce that (1) the extremal values of $u$ are only achieved at a vertex of the triangle, and (2) a generic acute triangle has exactly one (non-vertex) critical point and that each obtuse triangle has no (non-vertex) critical points. This settles the “hot spots” conjecture for triangles in the plane.
-
[Aronszajn] N. Aronszajn, "A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order," J. Math. Pures Appl. (9), vol. 36, pp. 235-249, 1957.
@ARTICLE{Aronszajn,
author = {Aronszajn, N.},
title = {A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order},
journal = {J. Math. Pures Appl. (9)},
fjournal = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
volume = {36},
year = {1957},
pages = {235--249},
issn = {0021-7824},
mrclass = {35.0X},
mrnumber = {0092067},
mrreviewer = {H. Bremekamp},
zblnumber = {0084.30402},
} -
[Atar-Burdzy]
R. Atar and K. Burdzy, "On Neumann eigenfunctions in lip domains," J. Amer. Math. Soc., vol. 17, iss. 2, pp. 243-265, 2004.@ARTICLE{Atar-Burdzy,
author = {Atar, Rami and Burdzy, Krzysztof},
title = {On {N}eumann eigenfunctions in lip domains},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the American Mathematical Society},
volume = {17},
year = {2004},
number = {2},
pages = {243--265},
issn = {0894-0347},
mrclass = {35J05 (35P05 60H20 60H30)},
mrnumber = {2051611},
mrreviewer = {Sergey G. Pyatkov},
doi = {10.1090/S0894-0347-04-00453-9},
url = {https://doi.org/10.1090/S0894-0347-04-00453-9},
zblnumber = {1151.35322},
} -
[Banuelos-Burdzy] R. Bañuelos and K. Burdzy, "On the “hot spots” conjecture of J. Rauch," J. Funct. Anal., vol. 164, iss. 1, pp. 1-33, 1999.
@ARTICLE{Banuelos-Burdzy,
author = {Ba\~{n}uelos, Rodrigo and Burdzy, Krzysztof},
title = {On the ``hot spots'' conjecture of {J}. {R}auch},
journal = {J. Funct. Anal.},
fjournal = {Journal of Functional Analysis},
volume = {164},
year = {1999},
number = {1},
pages = {1--33},
issn = {0022-1236},
mrclass = {35K05 (31C05 35B40 35B50 60J45)},
mrnumber = {1694534},
mrreviewer = {Zhongmin Qian},
doi = {10.1006/jfan.1999.3397},
url = {https://doi.org/10.1006/jfan.1999.3397},
zblnumber = {0938.35045},
} -
[Burdzy-Werner] K. Burdzy and W. Werner, "A counterexample to the “hot spots” conjecture," Ann. of Math. (2), vol. 149, iss. 1, pp. 309-317, 1999.
@ARTICLE{Burdzy-Werner,
author = {Burdzy, Krzysztof and Werner, Wendelin},
title = {A counterexample to the ``hot spots'' conjecture},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {149},
year = {1999},
number = {1},
pages = {309--317},
issn = {0003-486X},
mrclass = {35J25 (35P15)},
mrnumber = {1680567},
mrreviewer = {Vitaly A. Volpert},
doi = {10.2307/121027},
url = {https://doi.org/10.2307/121027},
zblnumber = {0919.35094},
} -
[Burdzy] K. Burdzy, "The hot spots problem in planar domains with one hole," Duke Math. J., vol. 129, iss. 3, pp. 481-502, 2005.
@ARTICLE{Burdzy,
author = {Burdzy, Krzysztof},
title = {The hot spots problem in planar domains with one hole},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {129},
year = {2005},
number = {3},
pages = {481--502},
issn = {0012-7094},
mrclass = {35J05 (35J25)},
mrnumber = {2169871},
mrreviewer = {Khairia E. Abd El-Fattah El-Nadi},
doi = {10.1215/S0012-7094-05-12932-5},
url = {https://doi.org/10.1215/S0012-7094-05-12932-5},
zblnumber = {1154.35330},
} -
[Cheng] S. Y. Cheng, "Eigenfunctions and nodal sets," Comment. Math. Helv., vol. 51, iss. 1, pp. 43-55, 1976.
@ARTICLE{Cheng,
author = {Cheng, Shiu Yuen},
title = {Eigenfunctions and nodal sets},
journal = {Comment. Math. Helv.},
fjournal = {Commentarii Mathematici Helvetici},
volume = {51},
year = {1976},
number = {1},
pages = {43--55},
issn = {0010-2571},
mrclass = {58G99 (35P15)},
mrnumber = {0397805},
mrreviewer = {Shukichi Tanno},
doi = {10.1007/BF02568142},
url = {https://doi.org/10.1007/BF02568142},
zblnumber = {0334.35022},
} -
[Coleman] N. E. Coleman, fepy, 2016.
@MISC{Coleman,
author = {Coleman, N. E.},
title = {{f}epy},
note = {Finite elements in python},
url = {https://github.com/necoleman/fepy},
year = {2016},
zblnumber = {},
} -
[Jerison-Nadirashvili] D. Jerison and N. Nadirashvili, "The “hot spots” conjecture for domains with two axes of symmetry," J. Amer. Math. Soc., vol. 13, iss. 4, pp. 741-772, 2000.
@ARTICLE{Jerison-Nadirashvili,
author = {Jerison, David and Nadirashvili, Nikolai},
title = {The ``hot spots'' conjecture for domains with two axes of symmetry},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the American Mathematical Society},
volume = {13},
year = {2000},
number = {4},
pages = {741--772},
issn = {0894-0347},
mrclass = {35J25 (35B65 35P15)},
mrnumber = {1775736},
mrreviewer = {Zhongmin Qian},
doi = {10.1090/S0894-0347-00-00346-5},
url = {https://doi.org/10.1090/S0894-0347-00-00346-5},
zblnumber = {0948.35029},
} -
[Kato] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.
@BOOK{Kato,
author = {Kato, Tosio},
title = {Perturbation Theory for Linear Operators},
series = {Classics Math.},
note = {reprint of the 1980 edition},
publisher = {Springer-Verlag, Berlin},
year = {1995},
pages = {xxii+619},
isbn = {3-540-58661-X},
mrclass = {47A55 (46-00 47-00)},
mrnumber = {1335452},
zblnumber = {0836.47009},
} -
[Kawohl] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Springer-Verlag, Berlin, 1985, vol. 1150.
@BOOK{Kawohl,
author = {Kawohl, Bernhard},
title = {Rearrangements and Convexity of Level Sets in {PDE}},
series = {Lecture Notes in Math.},
volume = {1150},
publisher = {Springer-Verlag, Berlin},
year = {1985},
pages = {iv+136},
isbn = {3-540-15693-3},
mrclass = {35-02 (35B50 35J60 49A50)},
mrnumber = {0810619},
mrreviewer = {Michael Wiegner},
doi = {10.1007/BFb0075060},
url = {https://doi.org/10.1007/BFb0075060},
zblnumber = {0593.35002},
} -
[Kuo] T. C. Kuo, "On $C^{0}$-sufficiency of jets of potential functions," Topology, vol. 8, pp. 167-171, 1969.
@ARTICLE{Kuo,
author = {Kuo, Tzee Char},
title = {On {$C\sp{0}$}-sufficiency of jets of potential functions},
journal = {Topology},
fjournal = {Topology. An International Journal of Mathematics},
volume = {8},
year = {1969},
pages = {167--171},
issn = {0040-9383},
mrclass = {57.20},
mrnumber = {0238338},
mrreviewer = {E. A. Feldman},
doi = {10.1016/0040-9383(69)90007-X},
url = {https://doi.org/10.1016/0040-9383(69)90007-X},
zblnumber = {0183.04601},
} -
[Lebedev] N. N. Lebedev, Special Functions and their Applications, Dover Publ., Inc., New York, 1972.
@BOOK{Lebedev,
author = {Lebedev, N. N.},
title = {Special Functions and their Applications},
note = {revised edition, translated from the Russian and edited by Richard A. Silverman; unabridged and corrected republication},
publisher = {Dover Publ., Inc., New York},
year = {1972},
pages = {xii+308},
mrclass = {33-XX (69.00)},
mrnumber = {0350075},
zblnumber = {0271.33001},
} -
[Lame] M. G. Lamé, Leçons sur le Théorie Mathématique de l’Elasticité des Corps SolidesBachelier, Paris, 1852.
@MISC{Lame,
author = {Lam{é},
M. G.},
title = {Le{ç}ons sur le Th{é}orie Math{é}matique de l'Elasticit{é} des Corps Solides},
publisher = {Bachelier, Paris},
year = {1852},
zblnumber = {},
} -
[Lojasiewicz] S. Łojasiewicz, "Sur le problème de la division," Studia Math., vol. 18, pp. 87-136, 1959.
@ARTICLE{Lojasiewicz,
author = {{\L}ojasiewicz, S.},
title = {Sur le problème de la division},
journal = {Studia Math.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
volume = {18},
year = {1959},
pages = {87--136},
issn = {0039-3223},
mrclass = {46.00},
mrnumber = {0107168},
mrreviewer = {L. Ehrenpreis},
doi = {10.4064/sm-18-1-87-136},
url = {https://doi.org/10.4064/sm-18-1-87-136},
zblnumber = {0115.10203},
} -
[Ltr-Rhl17] V. Lotoreichik and J. Rohleder, "Eigenvalue inequalities for the Laplacian with mixed boundary conditions," J. Differential Equations, vol. 263, iss. 1, pp. 491-508, 2017.
@ARTICLE{Ltr-Rhl17,
author = {Lotoreichik, Vladimir and Rohleder, Jonathan},
title = {Eigenvalue inequalities for the {L}aplacian with mixed boundary conditions},
journal = {J. Differential Equations},
fjournal = {Journal of Differential Equations},
volume = {263},
year = {2017},
number = {1},
pages = {491--508},
issn = {0022-0396},
mrclass = {35P15 (35J05)},
mrnumber = {3631314},
doi = {10.1016/j.jde.2017.02.043},
url = {https://doi.org/10.1016/j.jde.2017.02.043},
zblnumber = {1366.35106},
} -
[Miyamoto09] Y. Miyamoto, "The “hot spots” conjecture for a certain class of planar convex domains," J. Math. Phys., vol. 50, iss. 10, p. 103530, 2009.
@ARTICLE{Miyamoto09,
author = {Miyamoto, Yasuhito},
title = {The ``hot spots'' conjecture for a certain class of planar convex domains},
journal = {J. Math. Phys.},
fjournal = {Journal of Mathematical Physics},
volume = {50},
year = {2009},
number = {10},
pages = {103530, 7},
issn = {0022-2488},
mrclass = {35J05 (35P05)},
mrnumber = {2572703},
doi = {10.1063/1.3251335},
url = {https://doi.org/10.1063/1.3251335},
zblnumber = {1283.35016},
} -
[Miyamoto13] Y. Miyamoto, "A planar convex domain with many isolated “hot spots” on the boundary," Jpn. J. Ind. Appl. Math., vol. 30, iss. 1, pp. 145-164, 2013.
@ARTICLE{Miyamoto13,
author = {Miyamoto, Yasuhito},
title = {A planar convex domain with many isolated ``hot spots'' on the boundary},
journal = {Jpn. J. Ind. Appl. Math.},
fjournal = {Japan Journal of Industrial and Applied Mathematics},
volume = {30},
year = {2013},
number = {1},
pages = {145--164},
issn = {0916-7005},
mrclass = {35P15 (35J05)},
mrnumber = {3022811},
doi = {10.1007/s13160-012-0091-z},
url = {https://doi.org/10.1007/s13160-012-0091-z},
zblnumber = {1260.35090},
} -
[Nadir86] N. S. Nadirashvili, "Multiplicity of eigenvalues of the Neumann problem," Dokl. Akad. Nauk SSSR, vol. 286, iss. 6, pp. 1303-1305, 1986.
@ARTICLE{Nadir86,
author = {Nadirashvili, N. S.},
title = {Multiplicity of eigenvalues of the {N}eumann problem},
journal = {Dokl. Akad. Nauk SSSR},
fjournal = {Doklady Akademii Nauk SSSR},
volume = {286},
year = {1986},
number = {6},
pages = {1303--1305},
issn = {0002-3264},
mrclass = {35J25 (35P15 58G25)},
mrnumber = {0830292},
mrreviewer = {V. S. Rabinovich},
zblnumber = {0613.35052},
} -
[Otal-Rosas] J. Otal and E. Rosas, "Pour toute surface hyperbolique de genre $g$, $\lambda_{2g-2}>1/4$," Duke Math. J., vol. 150, iss. 1, pp. 101-115, 2009.
@ARTICLE{Otal-Rosas,
author = {Otal, Jean-Pierre and Rosas, Eulalio},
title = {Pour toute surface hyperbolique de genre {$g$, $\lambda_{2g-2}>1/4$}},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {150},
year = {2009},
number = {1},
pages = {101--115},
issn = {0012-7094},
mrclass = {58J50 (30F45 35J05 35P15)},
mrnumber = {2560109},
doi = {10.1215/00127094-2009-048},
url = {https://doi.org/10.1215/00127094-2009-048},
zblnumber = {1179.30041},
} -
[Pinsky] M. A. Pinsky, "The eigenvalues of an equilateral triangle," SIAM J. Math. Anal., vol. 11, iss. 5, pp. 819-827, 1980.
@ARTICLE{Pinsky,
author = {Pinsky, Mark A.},
title = {The eigenvalues of an equilateral triangle},
journal = {SIAM J. Math. Anal.},
fjournal = {SIAM Journal on Mathematical Analysis},
volume = {11},
year = {1980},
number = {5},
pages = {819--827},
issn = {0036-1410},
mrclass = {35P20 (12A05)},
mrnumber = {0586910},
mrreviewer = {Gerd Grubb},
doi = {10.1137/0511073},
url = {https://doi.org/10.1137/0511073},
zblnumber = {0462.35072},
} -
[Polya] G. Pólya, "Remarks on the foregoing paper," J. Math. Physics, vol. 31, pp. 55-57, 1952.
@ARTICLE{Polya,
author = {P{ó}lya, G.},
title = {Remarks on the foregoing paper},
journal = {J. Math. Physics},
volume = {31},
year = {1952},
pages = {55--57},
mrclass = {36.0X},
mrnumber = {0047237},
mrreviewer = {P. Funk},
doi = {10.1002/sapm195231155},
url = {https://doi.org/10.1002/sapm195231155},
zblnumber = {0046.32401},
} -
[Rauch] J. Rauch, "Five problems: An introduction to the qualitative theory of partial differential equations," in Partial Differential Equations and Related Topics, Springer, Berlin, 1975, vol. 446, pp. 355-369.
@incollection{Rauch,
author = {Rauch, Jeffrey},
title = {Five problems: {A}n introduction to the qualitative theory of partial differential equations},
booktitle = {Partial Differential Equations and Related Topics},
venue={{P}rogram, {T}ulane {U}niv., {N}ew {O}rleans, {L}a., 1974},
pages = {355--369},
series={Lecture Notes in Math.},
volume={446},
publisher = {Springer, Berlin},
year = {1975},
mrclass = {35BXX},
mrnumber = {0509045},
doi = {10.1007/BFb0070610},
url = {https://doi.org/10.1007/BFb0070610},
zblnumber = {0312.35001},
} -
[Polymath] relax Polymath, Polymath project 7 research thread 5: the hot spots conjecture.
@MISC{Polymath,
author={{\relax Polymath}},
title = {Polymath project 7 research thread 5: the hot spots conjecture},
note = {June 3, 2012 through August 9, 2013. \url{https://polymathprojects.org/2013/08/09/polymath7-research-thread-5-the-hot-spots-conjecture/}},
zblnumber = {},
} -
[Siudeja] B. Siudeja, "Hot spots conjecture for a class of acute triangles," Math. Z., vol. 280, iss. 3-4, pp. 783-806, 2015.
@ARTICLE{Siudeja,
author = {Siudeja, Bart{\l}omiej},
title = {Hot spots conjecture for a class of acute triangles},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {280},
year = {2015},
number = {3-4},
pages = {783--806},
issn = {0025-5874},
mrclass = {35P05 (35B38 35J05 35J25 35P15 58J50)},
mrnumber = {3369351},
mrreviewer = {En-Tao Zhao},
doi = {10.1007/s00209-015-1448-1},
url = {https://doi.org/10.1007/s00209-015-1448-1},
zblnumber = {1335.35164},
}
Full Article