American Institute of Mathematical Sciences

Advanced
July  2013, 33(7): 2939-2990. doi: 10.3934/dcds.2013.33.2939

The splitting lemmas for nonsmooth functionals on Hilbert spaces I

Guangcun Lu 1,

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received  August 2011 Revised  October 2012 Published  January 2013

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_\Omega f(x, u,\cdots, D^mu)dx$ as in (1.1). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincaré-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. Different from the usual implicit function theorem method and dynamical system one our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [19] with some techniques from [27,43,46]. Our theory is applicable to the Lagrangian systems on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.
Keywords: splitting lemma, Critical groups, shifting theorem..
Mathematics Subject Classification: Primary: 58E05, 49J52, 49J4.
Citation: Guangcun Lu. The splitting lemmas for nonsmooth functionals on Hilbert spaces I. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2939-2990. doi: 10.3934/dcds.2013.33.2939
References:
[1]

Adv. Nonlinear Stud., 9 (2009), 597-623.  Google Scholar

[2]

J. Funct. Anal., 186 (2001), 117-152. doi: 10.1006/jfan.2001.3789.  Google Scholar

[3]

Math. Z., 233 (2000), 655-677. doi: 10.1007/s002090050492.  Google Scholar

[4]

in "Handbook of Global Analysis" Elsevier Science Ltd, (2008), 41-73. doi: 10.1016/B978-044452833-9.50003-6.  Google Scholar

[5]

Ann. Sci. Math. Quebec, 22 (1998), 131-148.  Google Scholar

[6]

Topol. Methods Nonlinear Anal., 16 (2000), 279-306.  Google Scholar

[7]

Acad. Press, New York-London, 1977.  Google Scholar

[8]

C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.  Google Scholar

[9]

Bull. Amer. Math. Soc. (N.S), 9 (1983), 1-39. doi: 10.1090/S0273-0979-1983-15153-4.  Google Scholar

[10]

Univ. de Montreal, 97, 1985.  Google Scholar

[11]

Birkhäuser, 1993.  Google Scholar

[12]

Springer Monogaphs in Mathematics, Springer, 2005.  Google Scholar

[13]

C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441-446.  Google Scholar

[14]

Adv. Nonlinear Stud., 9 (2009), 679-699.  Google Scholar

[15]

Wiley, New York, 1983.  Google Scholar

[16]

Springer, New York, 1990.  Google Scholar

[17]

J. Math. Anal. Appl., 196 (1995), 1050-1072. doi: 10.1006/jmaa.1995.1460.  Google Scholar

[18]

Gauthier-Villars, 1963.  Google Scholar

[19]

Proc. Amer. Math. Soc., 135 (2007), 921-927. doi: 10.1090/S0002-9939-06-08662-X.  Google Scholar

[20]

Topological Methods in Nonlinear Analysis, 29 (2007), 35-68.  Google Scholar

[21]

J. Funct. Anal., 124 (1994), 1-39. doi: 10.1006/jfan.1994.1096.  Google Scholar

[22]

A.I.H.P. Analyse Non linéaire, 6 (1989), 321-330.  Google Scholar

[23]

Topology, 8 (1969), 361-369.  Google Scholar

[24]

in "Nonlinear functional analysis and its applications, Part 1" (Berkeley, Calif., 1983), Proc. Symp. Pure Math., 45, Part 1, Providence, RI, (1986), 501-509.  Google Scholar

[25]

Springer-Verlag, 1975.  Google Scholar

[26]

in "Recent Developments in Optimization Theory and Nonlinear Analysis" (Jerusalem, 1995), Contemp. Math., 204, Amer. Math. Soc., Providence, RI, (1997), 139-147. doi: 10.1090/conm/204/02627.  Google Scholar

[27]

Nonlinear Analysis, 36 (1999), 943-960. doi: 10.1016/S0362-546X(97)00701-3.  Google Scholar

[28]

John wiley & Sons. Ins. 1978.  Google Scholar

[29]

$2^{nd}$ edition, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4684-0265-0.  Google Scholar

[30]

J. Funct. Anal., 221 (2005), 439-455. doi: 10.1016/j.jfa.2004.09.010.  Google Scholar

[31]

J. Differential Equations, 244 (2008), 2498-2528. doi: 10.1016/j.jde.2008.02.021.  Google Scholar

[32]

J. Funct. Anal., 261 (2011), 542-589. doi: 10.1016/j.jfa.2009.01.001.  Google Scholar

[33]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces,, preprint, ().   Google Scholar

[34]

G. Lu, Some critical point theorems and applications,, preprint, ().   Google Scholar

[35]

G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds,, preprint, ().   Google Scholar

[36]

Bull. Soc. Math., Belgique (B), 37 (1985), 23-29.  Google Scholar

[37]

Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.  Google Scholar

[38]

Abstract and Applied Analysis, 5 (2000), 113-118. doi: 10.1155/S1085337500000245.  Google Scholar

[39]

J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939.  Google Scholar

[40]

Mathematical Surveys and Monographs 161, American Mathematical Society, Providence Rhode Island 2010.  Google Scholar

[41]

Academic Press, New York-London, 1971.  Google Scholar

[42]

Springer, Berlin, 2007. doi: 10.1007/978-3-540-71333-3.  Google Scholar

[43]

[in Russian], Naukova Dumka, Kiev 1973.  Google Scholar

[44]

Teubner, Leipzig, 1986.  Google Scholar

[45]

Math. Ann., 263 (1983), 303-312. doi: 10.1007/BF01457133.  Google Scholar

[46]

J. Sov. Math., 67 (1993), 2713-2811. doi: 10.1007/BF01455151.  Google Scholar

show all references

References:
[1]

Adv. Nonlinear Stud., 9 (2009), 597-623.  Google Scholar

[2]

J. Funct. Anal., 186 (2001), 117-152. doi: 10.1006/jfan.2001.3789.  Google Scholar

[3]

Math. Z., 233 (2000), 655-677. doi: 10.1007/s002090050492.  Google Scholar

[4]

in "Handbook of Global Analysis" Elsevier Science Ltd, (2008), 41-73. doi: 10.1016/B978-044452833-9.50003-6.  Google Scholar

[5]

Ann. Sci. Math. Quebec, 22 (1998), 131-148.  Google Scholar

[6]

Topol. Methods Nonlinear Anal., 16 (2000), 279-306.  Google Scholar

[7]

Acad. Press, New York-London, 1977.  Google Scholar

[8]

C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.  Google Scholar

[9]

Bull. Amer. Math. Soc. (N.S), 9 (1983), 1-39. doi: 10.1090/S0273-0979-1983-15153-4.  Google Scholar

[10]

Univ. de Montreal, 97, 1985.  Google Scholar

[11]

Birkhäuser, 1993.  Google Scholar

[12]

Springer Monogaphs in Mathematics, Springer, 2005.  Google Scholar

[13]

C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441-446.  Google Scholar

[14]

Adv. Nonlinear Stud., 9 (2009), 679-699.  Google Scholar

[15]

Wiley, New York, 1983.  Google Scholar

[16]

Springer, New York, 1990.  Google Scholar

[17]

J. Math. Anal. Appl., 196 (1995), 1050-1072. doi: 10.1006/jmaa.1995.1460.  Google Scholar

[18]

Gauthier-Villars, 1963.  Google Scholar

[19]

Proc. Amer. Math. Soc., 135 (2007), 921-927. doi: 10.1090/S0002-9939-06-08662-X.  Google Scholar

[20]

Topological Methods in Nonlinear Analysis, 29 (2007), 35-68.  Google Scholar

[21]

J. Funct. Anal., 124 (1994), 1-39. doi: 10.1006/jfan.1994.1096.  Google Scholar

[22]

A.I.H.P. Analyse Non linéaire, 6 (1989), 321-330.  Google Scholar

[23]

Topology, 8 (1969), 361-369.  Google Scholar

[24]

in "Nonlinear functional analysis and its applications, Part 1" (Berkeley, Calif., 1983), Proc. Symp. Pure Math., 45, Part 1, Providence, RI, (1986), 501-509.  Google Scholar

[25]

Springer-Verlag, 1975.  Google Scholar

[26]

in "Recent Developments in Optimization Theory and Nonlinear Analysis" (Jerusalem, 1995), Contemp. Math., 204, Amer. Math. Soc., Providence, RI, (1997), 139-147. doi: 10.1090/conm/204/02627.  Google Scholar

[27]

Nonlinear Analysis, 36 (1999), 943-960. doi: 10.1016/S0362-546X(97)00701-3.  Google Scholar

[28]

John wiley & Sons. Ins. 1978.  Google Scholar

[29]

$2^{nd}$ edition, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4684-0265-0.  Google Scholar

[30]

J. Funct. Anal., 221 (2005), 439-455. doi: 10.1016/j.jfa.2004.09.010.  Google Scholar

[31]

J. Differential Equations, 244 (2008), 2498-2528. doi: 10.1016/j.jde.2008.02.021.  Google Scholar

[32]

J. Funct. Anal., 261 (2011), 542-589. doi: 10.1016/j.jfa.2009.01.001.  Google Scholar

[33]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces,, preprint, ().   Google Scholar

[34]

G. Lu, Some critical point theorems and applications,, preprint, ().   Google Scholar

[35]

G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds,, preprint, ().   Google Scholar

[36]

Bull. Soc. Math., Belgique (B), 37 (1985), 23-29.  Google Scholar

[37]

Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.  Google Scholar

[38]

Abstract and Applied Analysis, 5 (2000), 113-118. doi: 10.1155/S1085337500000245.  Google Scholar

[39]

J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939.  Google Scholar

[40]

Mathematical Surveys and Monographs 161, American Mathematical Society, Providence Rhode Island 2010.  Google Scholar

[41]

Academic Press, New York-London, 1971.  Google Scholar

[42]

Springer, Berlin, 2007. doi: 10.1007/978-3-540-71333-3.  Google Scholar

[43]

[in Russian], Naukova Dumka, Kiev 1973.  Google Scholar

[44]

Teubner, Leipzig, 1986.  Google Scholar

[45]

Math. Ann., 263 (1983), 303-312. doi: 10.1007/BF01457133.  Google Scholar

[46]

J. Sov. Math., 67 (1993), 2713-2811. doi: 10.1007/BF01455151.  Google Scholar

[1]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374
[2]

V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511
[3]

Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences. Electronic Research Archive, , () : -. doi: 10.3934/era.2021025
[4]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[5]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027
[6]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016
[7]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141
[8]

Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021022
[9]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[10]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021014
[11]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019
[12]

Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021019
[13]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395
[14]

Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004
[15]

G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010
[16]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040
[17]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
[18]

Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391
[19]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[20]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences