# American Institute of Mathematical Sciences

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

 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.
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] Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073 [2] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [3] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [4] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [5] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276 [6] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [7] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [8] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [9] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [10] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [11] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [12] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [13] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

2019 Impact Factor: 1.338