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Classification of singular sets of solutions to elliptic equations

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    * Corresponding author 

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)

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  • 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.

    Mathematics Subject Classification: Primary: 35J15, 28A75; Secondary: 35B38.

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  • [1] G. Alessandrini, Critical points of solutions of elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa-Cl. Sci., 14 (1987), 229-256. 
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [9] H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183.  doi: 10.1007/BF01393691.
    [10] A. Enciso and D. Peralta-Salas, Critical points of Green's functions on complete manifolds, J. Differ. Geom., 92 (2012), 1-29. 
    [11] H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.
    [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.
    [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.
    [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.
    [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.
    [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.
    [17] R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J. Differ. Geom., 30 (1989), 505-522. 
    [18] R. HardtM. Hoffmann-OstenhofT. Hoffmann-Ostenhof and N. Nadirashvili, Critical sets of solutions to elliptic equations, J. Differ. Geom., 51 (1999), 359-373. 
    [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.
    [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. 
    [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.
    [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.
    [23] F. H. Lin and X. P. Yang, Geometric Measure Theory - An Introduction, Adv. Math., vol.1, Science Press/International Press, Beijing/Boston, 2002.
    [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.
    [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.
    [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. 
    [27] T. M. MacRobertSpherical Harmonics, An Elementary Treatise on Harmonic Functions with Applications, Pergamon Press, Oxford-New York-Toronto, 1967. 
    [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. 
    [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.
    [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. 
    [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.
    [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.
    [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.
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