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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 Dierential Equations, East China Normal University, Shanghai 200062, China |
References:
[1] |
C. Bär, Zero sets of solutions to semilinear elliptic systems of first order,, Invent. Math., 138 (1999), 183.
doi: 10.1007/s002220050346. |
[2] |
V. I. Bakhtin, The Weierstrass-Malgrange preparation theorem in the finitely smooth case,, Func. Anal. Appl., 24 (1990), 86.
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.
doi: 10.1007/s002200050598. |
[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. |
[5] |
S. Y. Cheng, Eigenfunctions and nodal sets,, Comment. Math. Helv., 51 (1976), 43.
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.
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.
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.
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.
doi: 10.1017/S0956792500001558. |
[10] |
H. Federer, Geometric Measure Theory,, Springer-Verlag, (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.
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.
doi: 10.2140/apde.2012.5.1133. |
[13] |
Q. Han, Singular sets of solutions to elliptic equations,, Indiana Univ. Math. J., 43 (1994), 983.
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.
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.
doi: 10.1007/BF02392214. |
[17] |
R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations,, J. Diff. Geom., 30 (1989), 505.
|
[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. |
[19] |
H. Hironaka, On the presentations of resolution data (Notes by T.T. Moh),, In: Algebraic Analysis, (1988), 135.
|
[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. |
[21] |
F.-H. Lin, Complexity of solutions of partial differential equations,, Handbook of geometric analysis, 7 (2008), 229.
|
[22] |
F.-H. Lin and X. P. Yang, Geometric Measure Theory: An Introduction,, Advanced Mathematics (Beijing/Boston), (2002).
|
[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. |
[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. |
[25] |
J. Milnor, Morse Theory,, Princeton University Press, (1963).
|
[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. |
[28] |
I. G. Petrovski'vi and O. A. Ole'vinik, On the topology of real algebraic surfaces,, Amer. Math. Soc. Translation, 1952 (1952).
|
[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. |
[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. |
[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.
|
[32] |
Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings,, Math. Ann., 264 (1983), 495.
doi: 10.1007/BF01456957. |
[33] |
Y. Yomdin, The set of zeroes of an "almost polynomial" function,, Proc. Amer. Math. Soc., 90 (1984), 538.
doi: 10.2307/2045026. |
[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. |
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. |
[2] |
V. I. Bakhtin, The Weierstrass-Malgrange preparation theorem in the finitely smooth case,, Func. Anal. Appl., 24 (1990), 86.
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.
doi: 10.1007/s002200050598. |
[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. |
[5] |
S. Y. Cheng, Eigenfunctions and nodal sets,, Comment. Math. Helv., 51 (1976), 43.
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.
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.
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.
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.
doi: 10.1017/S0956792500001558. |
[10] |
H. Federer, Geometric Measure Theory,, Springer-Verlag, (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.
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.
doi: 10.2140/apde.2012.5.1133. |
[13] |
Q. Han, Singular sets of solutions to elliptic equations,, Indiana Univ. Math. J., 43 (1994), 983.
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.
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.
doi: 10.1007/BF02392214. |
[17] |
R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations,, J. Diff. Geom., 30 (1989), 505.
|
[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. |
[19] |
H. Hironaka, On the presentations of resolution data (Notes by T.T. Moh),, In: Algebraic Analysis, (1988), 135.
|
[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. |
[21] |
F.-H. Lin, Complexity of solutions of partial differential equations,, Handbook of geometric analysis, 7 (2008), 229.
|
[22] |
F.-H. Lin and X. P. Yang, Geometric Measure Theory: An Introduction,, Advanced Mathematics (Beijing/Boston), (2002).
|
[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. |
[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. |
[25] |
J. Milnor, Morse Theory,, Princeton University Press, (1963).
|
[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. |
[28] |
I. G. Petrovski'vi and O. A. Ole'vinik, On the topology of real algebraic surfaces,, Amer. Math. Soc. Translation, 1952 (1952).
|
[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. |
[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. |
[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.
|
[32] |
Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings,, Math. Ann., 264 (1983), 495.
doi: 10.1007/BF01456957. |
[33] |
Y. Yomdin, The set of zeroes of an "almost polynomial" function,, Proc. Amer. Math. Soc., 90 (1984), 538.
doi: 10.2307/2045026. |
[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. |
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