June  2020, 19(6): 2949-2964. doi: 10.3934/cpaa.2020129

Classification of singular sets of solutions to elliptic equations

1. 

School of Statistics and Mathematics, Nanjing Audit University, Nanjing, Jiangsu, 211815, China

2. 

School of Science, Nanjing Forestry University, Nanjing, Jiangsu, 210037, China

3. 

School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, China

* Corresponding author

Received  December 2018 Revised  December 2019 Published  March 2020

Fund Project: The work is supported by National Natural Science Foundation of China (No.11401307, No.11401310, No.11771214) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_0321). The first author is fully supported by China Scholarship Council(CSC) for visiting Rutgers University(201806840122)

In this paper, we mainly investigate the classification of singular sets of solutions to elliptic equations. Firstly, we define the $ j $-symmetric singular set $ S^j(u) $ of solution $ u $, and show that the Hausdorff dimension of the $ j $-symmetric singular set $ S^j(u) $ is not more than $ j $. Then we prove the generalized $ \varepsilon $-regularity lemma for $ j $-symmetric homogeneous harmonic polynomial $ P $ with origin $ 0 $ as the isolated critical point in $ \mathbb{R}^{n-j} $, and by the generalized $ \varepsilon $-regularity lemma, we show the Hausdorff measure estimate of the $ j $-symmetric singular set $ S^j(u) $. Moreover, we study the geometric structure of interior singular points of solutions $ u $ in a planar bounded domain.

Citation: Haiyun Deng, Hairong Liu, Long Tian. Classification of singular sets of solutions to elliptic equations. Communications on Pure & Applied Analysis, 2020, 19 (6) : 2949-2964. doi: 10.3934/cpaa.2020129
References:
[1]

G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa-Cl. Sci., 14 (1987), 229-256.   Google Scholar

[2]

G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.  Google Scholar

[3]

S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2$^nd$ edition, Graduate Texts in Mathematics, Vol. 137, Springer, New York, 2001. doi: 10.1007/978-1-4757-8137-3.  Google Scholar

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J. CheegerA. Naber and D. Valtorta, Critical sets of elliptic equations, Comm. Pure Appl. Math., 68 (2015), 173-209.  doi: 10.1002/cpa.21518.  Google Scholar

[5]

H. Y. DengH. R. Liu and and L. Tian, Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions, J. Differ. Equ., 265 (2018), 4133-4157.  doi: 10.1016/j.jde.2018.05.031.  Google Scholar

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H. Y. DengH. R. Liu and and L. Tian, Critical points of solutions for the mean curvature equation in strictly convex and nonconvex domains, Israel J. Math., 233 (2019), 311-333.  doi: 10.1007/s11856-019-1906-2.  Google Scholar

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H. Y. DengH. R. Liu and L. Tian, Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions, J. Math. Anal. Appl., 477 (2019), 1072-1086.  doi: 10.1016/j.jmaa.2019.04.075.  Google Scholar

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H. Y. Deng, H. R. Liu and X. P. Yang, Critical points of solutions to a kind of linear elliptic equations in multiply connected domains, preprint, arXiv: 1811.04758. Google Scholar

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H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183.  doi: 10.1007/BF01393691.  Google Scholar

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A. Enciso and D. Peralta-Salas, Critical points of Green's functions on complete manifolds, J. Differ. Geom., 92 (2012), 1-29.   Google Scholar

[11]

H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.  Google Scholar

[12]

N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.  doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[13]

Q. Han, Singular sets of solutions to elliptic equations, Indiana Univ. Math. J., 43 (1994), 983-1002.  doi: 10.1512/iumj.1994.43.43043.  Google Scholar

[14]

Q. HanR. Hardt and F. H. Lin, Geometric measure of singular sets of elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1425-1443.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1425::AID-CPA8>3.3.CO;2-V.  Google Scholar

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Q. HanR. Hardt and F. H. Lin, Singular sets of higher order elliptic equations, Commun. Partial Differ. Equ., 28 (2003), 2045-2063.  doi: 10.1081/PDE-120025495.  Google Scholar

[16]

Q. Han and F. H. Lin, Nodal sets of solutions of elliptic differential equations, Unpublished manuscript, 2008. Available from: http://nd.edu/qhan/nodal.pdf. Google Scholar

[17]

R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differ. Geom., 30 (1989), 505-522.   Google Scholar

[18]

R. HardtM. Hoffmann-OstenhofT. Hoffmann-Ostenhof and N. Nadirashvili, Critical sets of solutions to elliptic equations, J. Differ. Geom., 51 (1999), 359-373.   Google Scholar

[19]

M. Hoffmann-OstenhofT. Hoffmann-Ostenhof and N. Nadirashvili, Critical sets of smooth solutions to elliptic equations in dimension 3, Indiana Univ. Math. J., 45 (1996), 15-37.  doi: 10.1512/iumj.1996.45.1957.  Google Scholar

[20]

J. Jung and S. Zelditch, Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution, J. Differ. Geom., 102 (2016), 37-66.   Google Scholar

[21]

B. Laurent, Critical sets of eigenfunctions of the Laplacian, J. Geom. Phys., 62 (2012), 2024-2037.  doi: 10.1016/j.geomphys.2012.05.006.  Google Scholar

[22]

F.H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44 (1991), 287-308.  doi: 10.1002/cpa.3160440303.  Google Scholar

[23]

F. H. Lin and X. P. Yang, Geometric Measure Theory - An Introduction, Adv. Math., vol.1, Science Press/International Press, Beijing/Boston, 2002.  Google Scholar

[24]

A. Logunov, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Ann. Math., 187 (2018), 221-239.  doi: 10.4007/annals.2018.187.1.4.  Google Scholar

[25]

A. Logunov, Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture, Ann. Math., 187 (2018), 241-262.  doi: 10.4007/annals.2018.187.1.5.  Google Scholar

[26]

A. Logunov and E. Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three, Operator Theory: Advances and Applications, 261 (2018), 333-344.   Google Scholar

[27] T. M. MacRobert, Spherical Harmonics, An Elementary Treatise on Harmonic Functions with Applications, Pergamon Press, Oxford-New York-Toronto, 1967.   Google Scholar
[28]

R. Magnanini, An introduction to the study of critical points of solutions of elliptic and parabolic equations, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 121-166.   Google Scholar

[29]

A. Naber and D. Valtorta, Volume estimates on the critical sets of solutions to elliptic PDEs, Comm. Pure Appl. Math., 70 (2017), 1835-1897.  doi: 10.1002/cpa.21708.  Google Scholar

[30]

S. Sakaguchi, Critical points of solutions to the obstacle problem in the plane, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 21 (1994), 157-173.   Google Scholar

[31]

C. D. Sogge and S. Zelditch, Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett., 18 (2011), 25-37.  doi: 10.4310/MRL.2011.v18.n1.a3.  Google Scholar

[32]

L. Tian and X. P. Yang, Measure estimates of nodal sets of bi-harmonic functions, J. Differ. Equ., 256 (2014), 558-576.  doi: 10.1016/j.jde.2013.09.012.  Google Scholar

[33]

S. Zelditch, Hausdorff measure of nodal sets of analytic Steklov eigenfunctions, Math. Res. Lett., 22 (2015), 1821-1842.  doi: 10.4310/MRL.2015.v22.n6.a15.  Google Scholar

show all references

References:
[1]

G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa-Cl. Sci., 14 (1987), 229-256.   Google Scholar

[2]

G. Alessandrini and R. Magnanini, Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (1994), 1259-1268.  doi: 10.1137/S0036141093249080.  Google Scholar

[3]

S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2$^nd$ edition, Graduate Texts in Mathematics, Vol. 137, Springer, New York, 2001. doi: 10.1007/978-1-4757-8137-3.  Google Scholar

[4]

J. CheegerA. Naber and D. Valtorta, Critical sets of elliptic equations, Comm. Pure Appl. Math., 68 (2015), 173-209.  doi: 10.1002/cpa.21518.  Google Scholar

[5]

H. Y. DengH. R. Liu and and L. Tian, Critical points of solutions to a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions, J. Differ. Equ., 265 (2018), 4133-4157.  doi: 10.1016/j.jde.2018.05.031.  Google Scholar

[6]

H. Y. DengH. R. Liu and and L. Tian, Critical points of solutions for the mean curvature equation in strictly convex and nonconvex domains, Israel J. Math., 233 (2019), 311-333.  doi: 10.1007/s11856-019-1906-2.  Google Scholar

[7]

H. Y. DengH. R. Liu and L. Tian, Uniqueness of critical points of solutions to the mean curvature equation with Neumann and Robin boundary conditions, J. Math. Anal. Appl., 477 (2019), 1072-1086.  doi: 10.1016/j.jmaa.2019.04.075.  Google Scholar

[8]

H. Y. Deng, H. R. Liu and X. P. Yang, Critical points of solutions to a kind of linear elliptic equations in multiply connected domains, preprint, arXiv: 1811.04758. Google Scholar

[9]

H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183.  doi: 10.1007/BF01393691.  Google Scholar

[10]

A. Enciso and D. Peralta-Salas, Critical points of Green's functions on complete manifolds, J. Differ. Geom., 92 (2012), 1-29.   Google Scholar

[11]

H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.  Google Scholar

[12]

N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268.  doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[13]

Q. Han, Singular sets of solutions to elliptic equations, Indiana Univ. Math. J., 43 (1994), 983-1002.  doi: 10.1512/iumj.1994.43.43043.  Google Scholar

[14]

Q. HanR. Hardt and F. H. Lin, Geometric measure of singular sets of elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1425-1443.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1425::AID-CPA8>3.3.CO;2-V.  Google Scholar

[15]

Q. HanR. Hardt and F. H. Lin, Singular sets of higher order elliptic equations, Commun. Partial Differ. Equ., 28 (2003), 2045-2063.  doi: 10.1081/PDE-120025495.  Google Scholar

[16]

Q. Han and F. H. Lin, Nodal sets of solutions of elliptic differential equations, Unpublished manuscript, 2008. Available from: http://nd.edu/qhan/nodal.pdf. Google Scholar

[17]

R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differ. Geom., 30 (1989), 505-522.   Google Scholar

[18]

R. HardtM. Hoffmann-OstenhofT. Hoffmann-Ostenhof and N. Nadirashvili, Critical sets of solutions to elliptic equations, J. Differ. Geom., 51 (1999), 359-373.   Google Scholar

[19]

M. Hoffmann-OstenhofT. Hoffmann-Ostenhof and N. Nadirashvili, Critical sets of smooth solutions to elliptic equations in dimension 3, Indiana Univ. Math. J., 45 (1996), 15-37.  doi: 10.1512/iumj.1996.45.1957.  Google Scholar

[20]

J. Jung and S. Zelditch, Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution, J. Differ. Geom., 102 (2016), 37-66.   Google Scholar

[21]

B. Laurent, Critical sets of eigenfunctions of the Laplacian, J. Geom. Phys., 62 (2012), 2024-2037.  doi: 10.1016/j.geomphys.2012.05.006.  Google Scholar

[22]

F.H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44 (1991), 287-308.  doi: 10.1002/cpa.3160440303.  Google Scholar

[23]

F. H. Lin and X. P. Yang, Geometric Measure Theory - An Introduction, Adv. Math., vol.1, Science Press/International Press, Beijing/Boston, 2002.  Google Scholar

[24]

A. Logunov, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Ann. Math., 187 (2018), 221-239.  doi: 10.4007/annals.2018.187.1.4.  Google Scholar

[25]

A. Logunov, Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture, Ann. Math., 187 (2018), 241-262.  doi: 10.4007/annals.2018.187.1.5.  Google Scholar

[26]

A. Logunov and E. Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three, Operator Theory: Advances and Applications, 261 (2018), 333-344.   Google Scholar

[27] T. M. MacRobert, Spherical Harmonics, An Elementary Treatise on Harmonic Functions with Applications, Pergamon Press, Oxford-New York-Toronto, 1967.   Google Scholar
[28]

R. Magnanini, An introduction to the study of critical points of solutions of elliptic and parabolic equations, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 121-166.   Google Scholar

[29]

A. Naber and D. Valtorta, Volume estimates on the critical sets of solutions to elliptic PDEs, Comm. Pure Appl. Math., 70 (2017), 1835-1897.  doi: 10.1002/cpa.21708.  Google Scholar

[30]

S. Sakaguchi, Critical points of solutions to the obstacle problem in the plane, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 21 (1994), 157-173.   Google Scholar

[31]

C. D. Sogge and S. Zelditch, Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett., 18 (2011), 25-37.  doi: 10.4310/MRL.2011.v18.n1.a3.  Google Scholar

[32]

L. Tian and X. P. Yang, Measure estimates of nodal sets of bi-harmonic functions, J. Differ. Equ., 256 (2014), 558-576.  doi: 10.1016/j.jde.2013.09.012.  Google Scholar

[33]

S. Zelditch, Hausdorff measure of nodal sets of analytic Steklov eigenfunctions, Math. Res. Lett., 22 (2015), 1821-1842.  doi: 10.4310/MRL.2015.v22.n6.a15.  Google Scholar

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