American Institute of Mathematical Sciences

Advanced
August  2016, 36(8): 4517-4529. doi: 10.3934/dcds.2016.36.4517

On the Betti numbers of level sets of solutions to elliptic equations

Fanghua Lin 1, and  Dan Liu 2,

1. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012
2. 

Center for Partial Di erential Equations, East China Normal University, Shanghai 200062, China

Received  May 2015 Revised  February 2016 Published  March 2016

In this paper we study the topological properties of the level sets, $ \S_{t}(u)=\left\{x:~u(x)= t \right\}$, of solutions $u$ of second order elliptic equations with vanishing zeroth order terms. We show that the total Betti number of level sets $\S_{t}$ is a uniformly bounded function of the parameter $t$. The uniform bound can be estimated in terms of the analytic coefficients as well as the generalized degrees of the corresponding solutions. Such an estimate is also valid for the nodal sets of solutions of the same type equations with zeroth order terms. In general, it is possible to derive from our analysis an estimate for the total Betti numbers of level sets, for large measure set of $t's$, when coefficients are sufficiently smooth, and therefore a $L^{p}$ bound on Betti numbers as a function of $t$. These estimates are obtained by a quantitative Stability Lemma and a quantitative Morse Lemma.
Keywords: elliptic PDEs., Betti number, level set, nodal set.
Mathematics Subject Classification: 82D55, 35B25, 35B40, 35Q5.
Citation: Fanghua Lin, Dan Liu. On the Betti numbers of level sets of solutions to elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4517-4529. doi: 10.3934/dcds.2016.36.4517
References:
[1]

C. Bär, Zero sets of solutions to semilinear elliptic systems of first order,, Invent. Math., 138 (1999), 183.  doi: 10.1007/s002220050346.  Google Scholar

[2]

V. I. Bakhtin, The Weierstrass-Malgrange preparation theorem in the finitely smooth case,, Func. Anal. Appl., 24 (1990), 86.  doi: 10.1007/BF01077701.  Google Scholar

[3]

J. Berger and J. Rubinstein, On the zero set of the wave function in superconductivity,, Comm. Math. Phys., 202 (1999), 621.  doi: 10.1007/s002200050598.  Google Scholar

[4]

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

[5]

S. Y. Cheng, Eigenfunctions and nodal sets,, Comment. Math. Helv., 51 (1976), 43.  doi: 10.1007/BF02568142.  Google Scholar

[6]

T. H. Colding and W. P. Minicozzi, Lower bounds for nodal sets of eigenfunctions,, Comm. Math. Phys., 306 (2011), 777.  doi: 10.1007/s00220-011-1225-x.  Google Scholar

[7]

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

[8]

H. Donnelly and C. Fefferman, Nodal sets for eigenfunctions of the Laplacian on surfaces,, J. Amer. Math. Soc., 3 (1990), 333.  doi: 10.1090/S0894-0347-1990-1035413-2.  Google Scholar

[9]

C. M. Elliott, H. Matano and Qi Tang, Zeros of a complex Ginzburg-Landau order parameter with applications to superconductivity,, European J. Appl. Math., 5 (1994), 431.  doi: 10.1017/S0956792500001558.  Google Scholar

[10]

H. Federer, Geometric Measure Theory,, Springer-Verlag, (1969).   Google Scholar

[11]

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

[12]

H. Hamid and C. Sogge, A natural lower bound for the size of nodal sets,, Anal. PDE, 5 (2012), 1133.  doi: 10.2140/apde.2012.5.1133.  Google Scholar

[13]

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

[14]

Q. Han, R. Hardt and F.-H. Lin, Geometric measure of singular sets of elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1425.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1425::AID-CPA8>3.0.CO;2-3.  Google Scholar

[15]

R. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps,, Invent. Math., 38 (): 207.  doi: 10.1007/BF01403128.  Google Scholar

[16]

R. Hardt, Slicing and intersection theory for chains associated with real analytic varieties,, Acta Math., 129 (1972), 75.  doi: 10.1007/BF02392214.  Google Scholar

[17]

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

[18]

B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in nonsimple connected domains,, Comm. Math. Phys., 202 (1999), 629.  doi: 10.1007/s002200050599.  Google Scholar

[19]

H. Hironaka, On the presentations of resolution data (Notes by T.T. Moh),, In: Algebraic Analysis, (1988), 135.   Google Scholar

[20]

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

[21]

F.-H. Lin, Complexity of solutions of partial differential equations,, Handbook of geometric analysis, 7 (2008), 229.   Google Scholar

[22]

F.-H. Lin and X. P. Yang, Geometric Measure Theory: An Introduction,, Advanced Mathematics (Beijing/Boston), (2002).   Google Scholar

[23]

D. Liu, Hausdorff measure of critical sets of solutions to magnetic Schrödinger equations,, Cal. Var. PDEs., 41 (2011), 179.  doi: 10.1007/s00526-010-0358-7.  Google Scholar

[24]

J. Milnor, On the Betti numbers of real varieties,, Proc. Amer. Math. Soc., 15 (1964), 275.  doi: 10.1090/S0002-9939-1964-0161339-9.  Google Scholar

[25]

J. Milnor, Morse Theory,, Princeton University Press, (1963).   Google Scholar

[26]

A. Naber and D. Valtorta, Volume estimates on critical sets of elliptic PDEs,, , ().   Google Scholar

[27]

X. B. Pan, Nodal sets of solutions of equations involving magnetic Schrödinger operator in 3-dimensions,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2738752.  Google Scholar

[28]

I. G. Petrovski'vi and O. A. Ole'vinik, On the topology of real algebraic surfaces,, Amer. Math. Soc. Translation, 1952 (1952).   Google Scholar

[29]

A. Sard, The measure of the critical values of differentiable maps,, Bull. Amer. Math. Soc., 48 (1942), 883.  doi: 10.1090/S0002-9904-1942-07811-6.  Google Scholar

[30]

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

[31]

R. Thom, Sur l'homologie des vari'et'es alg'ebriques r'eelles,, (French) 1965 Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) pp. 255-265, (1965), 255.   Google Scholar

[32]

Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings,, Math. Ann., 264 (1983), 495.  doi: 10.1007/BF01456957.  Google Scholar

[33]

Y. Yomdin, The set of zeroes of an "almost polynomial" function,, Proc. Amer. Math. Soc., 90 (1984), 538.  doi: 10.2307/2045026.  Google Scholar

[34]

Y. Yomdin, Global bounds for the Betti numbers of regular fibers of differentiable mappings,, Topology, 24 (1985), 145.  doi: 10.1016/0040-9383(85)90051-5.  Google Scholar

show all references

References:
[1]

C. Bär, Zero sets of solutions to semilinear elliptic systems of first order,, Invent. Math., 138 (1999), 183.  doi: 10.1007/s002220050346.  Google Scholar

[2]

V. I. Bakhtin, The Weierstrass-Malgrange preparation theorem in the finitely smooth case,, Func. Anal. Appl., 24 (1990), 86.  doi: 10.1007/BF01077701.  Google Scholar

[3]

J. Berger and J. Rubinstein, On the zero set of the wave function in superconductivity,, Comm. Math. Phys., 202 (1999), 621.  doi: 10.1007/s002200050598.  Google Scholar

[4]

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

[5]

S. Y. Cheng, Eigenfunctions and nodal sets,, Comment. Math. Helv., 51 (1976), 43.  doi: 10.1007/BF02568142.  Google Scholar

[6]

T. H. Colding and W. P. Minicozzi, Lower bounds for nodal sets of eigenfunctions,, Comm. Math. Phys., 306 (2011), 777.  doi: 10.1007/s00220-011-1225-x.  Google Scholar

[7]

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

[8]

H. Donnelly and C. Fefferman, Nodal sets for eigenfunctions of the Laplacian on surfaces,, J. Amer. Math. Soc., 3 (1990), 333.  doi: 10.1090/S0894-0347-1990-1035413-2.  Google Scholar

[9]

C. M. Elliott, H. Matano and Qi Tang, Zeros of a complex Ginzburg-Landau order parameter with applications to superconductivity,, European J. Appl. Math., 5 (1994), 431.  doi: 10.1017/S0956792500001558.  Google Scholar

[10]

H. Federer, Geometric Measure Theory,, Springer-Verlag, (1969).   Google Scholar

[11]

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

[12]

H. Hamid and C. Sogge, A natural lower bound for the size of nodal sets,, Anal. PDE, 5 (2012), 1133.  doi: 10.2140/apde.2012.5.1133.  Google Scholar

[13]

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

[14]

Q. Han, R. Hardt and F.-H. Lin, Geometric measure of singular sets of elliptic equations,, Comm. Pure Appl. Math., 51 (1998), 1425.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1425::AID-CPA8>3.0.CO;2-3.  Google Scholar

[15]

R. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps,, Invent. Math., 38 (): 207.  doi: 10.1007/BF01403128.  Google Scholar

[16]

R. Hardt, Slicing and intersection theory for chains associated with real analytic varieties,, Acta Math., 129 (1972), 75.  doi: 10.1007/BF02392214.  Google Scholar

[17]

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

[18]

B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and M. Owen, Nodal sets for groundstates of Schrödinger operators with zero magnetic field in nonsimple connected domains,, Comm. Math. Phys., 202 (1999), 629.  doi: 10.1007/s002200050599.  Google Scholar

[19]

H. Hironaka, On the presentations of resolution data (Notes by T.T. Moh),, In: Algebraic Analysis, (1988), 135.   Google Scholar

[20]

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

[21]

F.-H. Lin, Complexity of solutions of partial differential equations,, Handbook of geometric analysis, 7 (2008), 229.   Google Scholar

[22]

F.-H. Lin and X. P. Yang, Geometric Measure Theory: An Introduction,, Advanced Mathematics (Beijing/Boston), (2002).   Google Scholar

[23]

D. Liu, Hausdorff measure of critical sets of solutions to magnetic Schrödinger equations,, Cal. Var. PDEs., 41 (2011), 179.  doi: 10.1007/s00526-010-0358-7.  Google Scholar

[24]

J. Milnor, On the Betti numbers of real varieties,, Proc. Amer. Math. Soc., 15 (1964), 275.  doi: 10.1090/S0002-9939-1964-0161339-9.  Google Scholar

[25]

J. Milnor, Morse Theory,, Princeton University Press, (1963).   Google Scholar

[26]

A. Naber and D. Valtorta, Volume estimates on critical sets of elliptic PDEs,, , ().   Google Scholar

[27]

X. B. Pan, Nodal sets of solutions of equations involving magnetic Schrödinger operator in 3-dimensions,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2738752.  Google Scholar

[28]

I. G. Petrovski'vi and O. A. Ole'vinik, On the topology of real algebraic surfaces,, Amer. Math. Soc. Translation, 1952 (1952).   Google Scholar

[29]

A. Sard, The measure of the critical values of differentiable maps,, Bull. Amer. Math. Soc., 48 (1942), 883.  doi: 10.1090/S0002-9904-1942-07811-6.  Google Scholar

[30]

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

[31]

R. Thom, Sur l'homologie des vari'et'es alg'ebriques r'eelles,, (French) 1965 Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) pp. 255-265, (1965), 255.   Google Scholar

[32]

Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings,, Math. Ann., 264 (1983), 495.  doi: 10.1007/BF01456957.  Google Scholar

[33]

Y. Yomdin, The set of zeroes of an "almost polynomial" function,, Proc. Amer. Math. Soc., 90 (1984), 538.  doi: 10.2307/2045026.  Google Scholar

[34]

Y. Yomdin, Global bounds for the Betti numbers of regular fibers of differentiable mappings,, Topology, 24 (1985), 145.  doi: 10.1016/0040-9383(85)90051-5.  Google Scholar

[1]

Peter Poláčik. On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2657-2667. doi: 10.3934/dcds.2014.34.2657
[2]

Bin Dong, Aichi Chien, Yu Mao, Jian Ye, Fernando Vinuela, Stanley Osher. Level set based brain aneurysm capturing in 3D. Inverse Problems & Imaging, 2010, 4 (2) : 241-255. doi: 10.3934/ipi.2010.4.241
[3]

Federico Rodriguez Hertz, Jana Rodriguez Hertz. Cohomology free systems and the first Betti number. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 193-196. doi: 10.3934/dcds.2006.15.193
[4]

Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115
[5]

Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems & Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459
[6]

Esther Klann, Ronny Ramlau, Wolfgang Ring. A Mumford-Shah level-set approach for the inversion and segmentation of SPECT/CT data. Inverse Problems & Imaging, 2011, 5 (1) : 137-166. doi: 10.3934/ipi.2011.5.137
[7]

Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems & Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479
[8]

Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019228
[9]

Lan Wen. On the preperiodic set. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237
[10]

David Auger, Irène Charon, Iiro Honkala, Olivier Hudry, Antoine Lobstein. Edge number, minimum degree, maximum independent set, radius and diameter in twin-free graphs. Advances in Mathematics of Communications, 2009, 3 (1) : 97-114. doi: 10.3934/amc.2009.3.97
[11]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[12]

Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227
[13]

Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
[14]

Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245
[15]

Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215
[16]

Humberto Ramos Quoirin, Kenichiro Umezu. A loop type component in the non-negative solutions set of an indefinite elliptic problem. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1255-1269. doi: 10.3934/cpaa.2018060
[17]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243
[18]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047
[19]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
[20]

Sanjit Chatterjee, Chethan Kamath, Vikas Kumar. Private set-intersection with common set-up. Advances in Mathematics of Communications, 2018, 12 (1) : 17-47. doi: 10.3934/amc.2018002

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences