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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 |
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.
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.
|
[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. Cheeger, A. 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. Deng, H. 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. Deng, H. 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. Deng, H. 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. Han, R. 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. Han, R. 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. Hardt, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and N. Nadirashvili,
Critical sets of solutions to elliptic equations, J. Differ. Geom., 51 (1999), 359-373.
|
[19] |
M. Hoffmann-Ostenhof, T. 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. MacRobert, Spherical 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. |
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.
|
[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. Cheeger, A. 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. Deng, H. 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. Deng, H. 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. Deng, H. 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. Han, R. 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. Han, R. 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. Hardt, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and N. Nadirashvili,
Critical sets of solutions to elliptic equations, J. Differ. Geom., 51 (1999), 359-373.
|
[19] |
M. Hoffmann-Ostenhof, T. 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. MacRobert, Spherical 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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