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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 and Continuous Dynamical Systems, 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), 183-202. doi: 10.1007/s002220050346.

[2]

V. I. Bakhtin, The Weierstrass-Malgrange preparation theorem in the finitely smooth case, Func. Anal. Appl., 24 (1990), 86-96. doi: 10.1007/BF01077701.

[3]

J. Berger and J. Rubinstein, On the zero set of the wave function in superconductivity, Comm. Math. Phys., 202 (1999), 621-628. doi: 10.1007/s002200050598.

[4]

J. Cheeger, A. Naber and D. Valtorta, Critical sets of elliptic equations, Comm. Pure Appl. Math., 68 (2015), 173-209, arXiv:1207.4236. doi: 10.1002/cpa.21518.

[5]

S. Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. doi: 10.1007/BF02568142.

[6]

T. H. Colding and W. P. Minicozzi, Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys., 306 (2011), 777-784. doi: 10.1007/s00220-011-1225-x.

[7]

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

[8]

H. Donnelly and C. Fefferman, Nodal sets for eigenfunctions of the Laplacian on surfaces, J. Amer. Math. Soc., 3 (1990), 333-353. doi: 10.1090/S0894-0347-1990-1035413-2.

[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), 431-448. doi: 10.1017/S0956792500001558.

[10]

H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969.

[11]

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

[12]

H. Hamid and C. Sogge, A natural lower bound for the size of nodal sets, Anal. PDE, 5 (2012), 1133-1137. doi: 10.2140/apde.2012.5.1133.

[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>3.0.CO;2-3.

[15]

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

[16]

R. Hardt, Slicing and intersection theory for chains associated with real analytic varieties, Acta Math., 129 (1972), 75-136. doi: 10.1007/BF02392214.

[17]

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

[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), 629-649. doi: 10.1007/s002200050599.

[19]

H. Hironaka, On the presentations of resolution data (Notes by T.T. Moh), In: Algebraic Analysis, Geometry and Number Theory 1988 (ed.J.I.Igusa), The Johns Hopkins University Press, 1989, 135-151.

[20]

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.

[21]

F.-H. Lin, Complexity of solutions of partial differential equations, Handbook of geometric analysis, 229-258, Adv. Lect. Math., 7, Int. Press, Someerville, MA, 2008.

[22]

F.-H. Lin and X. P. Yang, Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), 1, Science Press, Beijing, International Press, Boston, MA, 2002.

[23]

D. Liu, Hausdorff measure of critical sets of solutions to magnetic Schrödinger equations, Cal. Var. PDEs., 41 (2011), 179-202. doi: 10.1007/s00526-010-0358-7.

[24]

J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc., 15 (1964), 275-280. doi: 10.1090/S0002-9939-1964-0161339-9.

[25]

J. Milnor, Morse Theory, Princeton University Press, Princeton, N.J., 1963.

[26]

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

[27]

X. B. Pan, Nodal sets of solutions of equations involving magnetic Schrödinger operator in 3-dimensions, J. Math. Phys., 48 (2007), 053521, 20 pp. doi: 10.1063/1.2738752.

[28]

I. G. Petrovski'vi and O. A. Ole'vinik, On the topology of real algebraic surfaces, Amer. Math. Soc. Translation, 1952 (1952), 20pp.

[29]

A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883-890. doi: 10.1090/S0002-9904-1942-07811-6.

[30]

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.

[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, Princeton Univ. Press, Princeton, N.J.

[32]

Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings, Math. Ann., 264 (1983), 495-515. doi: 10.1007/BF01456957.

[33]

Y. Yomdin, The set of zeroes of an "almost polynomial" function, Proc. Amer. Math. Soc., 90 (1984), 538-542. doi: 10.2307/2045026.

[34]

Y. Yomdin, Global bounds for the Betti numbers of regular fibers of differentiable mappings, Topology, 24 (1985), 145-152. doi: 10.1016/0040-9383(85)90051-5.

show all references

References:
[1]

C. Bär, Zero sets of solutions to semilinear elliptic systems of first order, Invent. Math., 138 (1999), 183-202. doi: 10.1007/s002220050346.

[2]

V. I. Bakhtin, The Weierstrass-Malgrange preparation theorem in the finitely smooth case, Func. Anal. Appl., 24 (1990), 86-96. doi: 10.1007/BF01077701.

[3]

J. Berger and J. Rubinstein, On the zero set of the wave function in superconductivity, Comm. Math. Phys., 202 (1999), 621-628. doi: 10.1007/s002200050598.

[4]

J. Cheeger, A. Naber and D. Valtorta, Critical sets of elliptic equations, Comm. Pure Appl. Math., 68 (2015), 173-209, arXiv:1207.4236. doi: 10.1002/cpa.21518.

[5]

S. Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1976), 43-55. doi: 10.1007/BF02568142.

[6]

T. H. Colding and W. P. Minicozzi, Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys., 306 (2011), 777-784. doi: 10.1007/s00220-011-1225-x.

[7]

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

[8]

H. Donnelly and C. Fefferman, Nodal sets for eigenfunctions of the Laplacian on surfaces, J. Amer. Math. Soc., 3 (1990), 333-353. doi: 10.1090/S0894-0347-1990-1035413-2.

[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), 431-448. doi: 10.1017/S0956792500001558.

[10]

H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969.

[11]

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

[12]

H. Hamid and C. Sogge, A natural lower bound for the size of nodal sets, Anal. PDE, 5 (2012), 1133-1137. doi: 10.2140/apde.2012.5.1133.

[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>3.0.CO;2-3.

[15]

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

[16]

R. Hardt, Slicing and intersection theory for chains associated with real analytic varieties, Acta Math., 129 (1972), 75-136. doi: 10.1007/BF02392214.

[17]

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

[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), 629-649. doi: 10.1007/s002200050599.

[19]

H. Hironaka, On the presentations of resolution data (Notes by T.T. Moh), In: Algebraic Analysis, Geometry and Number Theory 1988 (ed.J.I.Igusa), The Johns Hopkins University Press, 1989, 135-151.

[20]

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.

[21]

F.-H. Lin, Complexity of solutions of partial differential equations, Handbook of geometric analysis, 229-258, Adv. Lect. Math., 7, Int. Press, Someerville, MA, 2008.

[22]

F.-H. Lin and X. P. Yang, Geometric Measure Theory: An Introduction, Advanced Mathematics (Beijing/Boston), 1, Science Press, Beijing, International Press, Boston, MA, 2002.

[23]

D. Liu, Hausdorff measure of critical sets of solutions to magnetic Schrödinger equations, Cal. Var. PDEs., 41 (2011), 179-202. doi: 10.1007/s00526-010-0358-7.

[24]

J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc., 15 (1964), 275-280. doi: 10.1090/S0002-9939-1964-0161339-9.

[25]

J. Milnor, Morse Theory, Princeton University Press, Princeton, N.J., 1963.

[26]

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

[27]

X. B. Pan, Nodal sets of solutions of equations involving magnetic Schrödinger operator in 3-dimensions, J. Math. Phys., 48 (2007), 053521, 20 pp. doi: 10.1063/1.2738752.

[28]

I. G. Petrovski'vi and O. A. Ole'vinik, On the topology of real algebraic surfaces, Amer. Math. Soc. Translation, 1952 (1952), 20pp.

[29]

A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883-890. doi: 10.1090/S0002-9904-1942-07811-6.

[30]

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.

[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, Princeton Univ. Press, Princeton, N.J.

[32]

Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings, Math. Ann., 264 (1983), 495-515. doi: 10.1007/BF01456957.

[33]

Y. Yomdin, The set of zeroes of an "almost polynomial" function, Proc. Amer. Math. Soc., 90 (1984), 538-542. doi: 10.2307/2045026.

[34]

Y. Yomdin, Global bounds for the Betti numbers of regular fibers of differentiable mappings, Topology, 24 (1985), 145-152. doi: 10.1016/0040-9383(85)90051-5.

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