The splitting lemmas for nonsmooth functionals on Hilbert spaces I

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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

Abstract
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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 and Continuous Dynamical Systems, 2013, 33 (7) : 2939-2990. doi: 10.3934/dcds.2013.33.2939

References:

[1]	A. Abbondandolo and M. Schwarz, A smooth pseudo-gradient for the Lagrangian action functional, Adv. Nonlinear Stud., 9 (2009), 597-623.
[2]	T. Bartsch, Critical point theory on partially ordered hilbert spaces, J. Funct. Anal., 186 (2001), 117-152. doi: 10.1006/jfan.2001.3789.
[3]	T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233 (2000), 655-677. doi: 10.1007/s002090050492.
[4]	T. Bartsch, A. Szulkin and M. Willem, Morse theory and nonlinear differential equations, in "Handbook of Global Analysis" Elsevier Science Ltd, (2008), 41-73. doi: 10.1016/B978-044452833-9.50003-6.
[5]	P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree, Ann. Sci. Math. Quebec, 22 (1998), 131-148.
[6]	P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds, Topol. Methods Nonlinear Anal., 16 (2000), 279-306.
[7]	M. Berger, "Nonlinearity and Functional Analysis," Acad. Press, New York-London, 1977.
[8]	H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.
[9]	F. E. Browder, Fixed point theory and nonlinear problem, Bull. Amer. Math. Soc. (N.S), 9 (1983), 1-39. doi: 10.1090/S0273-0979-1983-15153-4.
[10]	K. C. Chang, "Infinite Dimensional Morse Theory and its Applications," Univ. de Montreal, 97, 1985.
[11]	K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problem," Birkhäuser, 1993.
[12]	K. C. Chang, "Methods in Nonlinear Analysis," Springer Monogaphs in Mathematics, Springer, 2005.
[13]	K. C. Chang, $H^1$ versus $C^1$ isolated critical points, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441-446.
[14]	S. Cingolani and M. Degiovanni, On the Poincaré-Hopf theorem for functionals defined on Banach spaces, Adv. Nonlinear Stud., 9 (2009), 679-699.
[15]	F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983.
[16]	J. B. Conway, "A Course in Functional Analysis," Springer, New York, 1990.
[17]	J. N. Corvellec, Morse theory for continuous functionals, J. Math. Anal. Appl., 196 (1995), 1050-1072. doi: 10.1006/jmaa.1995.1460.
[18]	J. Dieudonné, "Fondements de L'Analyse Moderne," Gauthier-Villars, 1963.
[19]	D. M. Duc, T. V. Hung and N. T. Khai, Morse-Palais lemma for nonsmooth functionals on normed spaces, Proc. Amer. Math. Soc., 135 (2007), 921-927. doi: 10.1090/S0002-9939-06-08662-X.
[20]	D. M. Duc, T. V. Hung and N. T. Khai, Critical points of non-$C^2$ functionals, Topological Methods in Nonlinear Analysis, 29 (2007), 35-68.
[21]	P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier, Orientability of fredholm families and topological degree for orientable nonlinear fredholm mappings, J. Funct. Anal., 124 (1994), 1-39. doi: 10.1006/jfan.1994.1096.
[22]	N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, A.I.H.P. Analyse Non linéaire, 6 (1989), 321-330.
[23]	D. Gromoll and W. Meyer, On differentiable functions with isolated critical points, Topology, 8 (1969), 361-369.
[24]	H. Hofer, The topological degree at a critical point of mountain pass type, in "Nonlinear functional analysis and its applications, Part 1" (Berkeley, Calif., 1983), Proc. Symp. Pure Math., 45, Part 1, Providence, RI, (1986), 501-509.
[25]	D. Husemoller, "Fibre Bundle," Springer-Verlag, 1975.
[26]	A. Ioffe and E. Schwartzman, Parametric Morse lemmas for $C^{1,1}$-functions, 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.
[27]	M. Jiang, A generalization of Morse lemma and its applications, Nonlinear Analysis, 36 (1999), 943-960. doi: 10.1016/S0362-546X(97)00701-3.
[28]	E. Kreyszig, "Introduction Functional Analysis with Applications," John wiley & Sons. Ins. 1978.
[29]	S. Lang, "Differential Manifolds," $2^{nd}$ edition, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4684-0265-0.
[30]	C. Li, S. -J. Li and J. Liu, Splitting theorem, Poincare-Hopf theorem and jumping nonlinear problems, J. Funct. Anal., 221 (2005), 439-455. doi: 10.1016/j.jfa.2004.09.010.
[31]	C. Li, S.-J. Li, Z. Liu and J. Pan, On the Fucík spectrum, J. Differential Equations, 244 (2008), 2498-2528. doi: 10.1016/j.jde.2008.02.021.
[32]	G. Lu, Corrigendum to "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems" [J. Funct. Anal. 256(9)(2009)2967-3034], J. Funct. Anal., 261 (2011), 542-589. doi: 10.1016/j.jfa.2009.01.001.
[33]	G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces, preprint, arXiv:1102.2062.
[34]	G. Lu, Some critical point theorems and applications, preprint, arXiv:1102.2136.
[35]	G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds, preprint, arXiv:1212.2078.
[36]	Jean Mawhin and Michel Willem, On the generalized Morse Lemma, Bull. Soc. Math., Belgique (B), 37 (1985), 23-29.
[37]	Jean Mawhin and Michel Willem, "Critical Point Theory and Hamiltonian Systems," Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.
[38]	A. A. Moura and F. M. de Souza, A Morse lemma for degenerate critical points with low differentiability, Abstract and Applied Analysis, 5 (2000), 113-118. doi: 10.1155/S1085337500000245.
[39]	J. Pejsachowicz and P. R. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$, J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939.
[40]	K. Perera, R. P. Agarwal and Donal O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators," Mathematical Surveys and Monographs 161, American Mathematical Society, Providence Rhode Island 2010.
[41]	M. Schechter, "Principles of Functional Analysis," Academic Press, New York-London, 1971.
[42]	W. Schirotzek, "Nonsmooth Analysis," Springer, Berlin, 2007. doi: 10.1007/978-3-540-71333-3.
[43]	I. V. Skrypnik, "Nonlinear Elliptic Equations of a Higher Order," [in Russian], Naukova Dumka, Kiev 1973.
[44]	I. V. Skrypnik, "Nonlinear Elliptic Boundary Value Problems," Teubner, Leipzig, 1986.
[45]	A. Tromba, A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem, Math. Ann., 263 (1983), 303-312. doi: 10.1007/BF01457133.
[46]	S. A. Vakhrameev, Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems, J. Sov. Math., 67 (1993), 2713-2811. doi: 10.1007/BF01455151.

show all references

References:

[1]	A. Abbondandolo and M. Schwarz, A smooth pseudo-gradient for the Lagrangian action functional, Adv. Nonlinear Stud., 9 (2009), 597-623.
[2]	T. Bartsch, Critical point theory on partially ordered hilbert spaces, J. Funct. Anal., 186 (2001), 117-152. doi: 10.1006/jfan.2001.3789.
[3]	T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z., 233 (2000), 655-677. doi: 10.1007/s002090050492.
[4]	T. Bartsch, A. Szulkin and M. Willem, Morse theory and nonlinear differential equations, in "Handbook of Global Analysis" Elsevier Science Ltd, (2008), 41-73. doi: 10.1016/B978-044452833-9.50003-6.
[5]	P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree, Ann. Sci. Math. Quebec, 22 (1998), 131-148.
[6]	P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds, Topol. Methods Nonlinear Anal., 16 (2000), 279-306.
[7]	M. Berger, "Nonlinearity and Functional Analysis," Acad. Press, New York-London, 1977.
[8]	H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.
[9]	F. E. Browder, Fixed point theory and nonlinear problem, Bull. Amer. Math. Soc. (N.S), 9 (1983), 1-39. doi: 10.1090/S0273-0979-1983-15153-4.
[10]	K. C. Chang, "Infinite Dimensional Morse Theory and its Applications," Univ. de Montreal, 97, 1985.
[11]	K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problem," Birkhäuser, 1993.
[12]	K. C. Chang, "Methods in Nonlinear Analysis," Springer Monogaphs in Mathematics, Springer, 2005.
[13]	K. C. Chang, $H^1$ versus $C^1$ isolated critical points, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441-446.
[14]	S. Cingolani and M. Degiovanni, On the Poincaré-Hopf theorem for functionals defined on Banach spaces, Adv. Nonlinear Stud., 9 (2009), 679-699.
[15]	F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983.
[16]	J. B. Conway, "A Course in Functional Analysis," Springer, New York, 1990.
[17]	J. N. Corvellec, Morse theory for continuous functionals, J. Math. Anal. Appl., 196 (1995), 1050-1072. doi: 10.1006/jmaa.1995.1460.
[18]	J. Dieudonné, "Fondements de L'Analyse Moderne," Gauthier-Villars, 1963.
[19]	D. M. Duc, T. V. Hung and N. T. Khai, Morse-Palais lemma for nonsmooth functionals on normed spaces, Proc. Amer. Math. Soc., 135 (2007), 921-927. doi: 10.1090/S0002-9939-06-08662-X.
[20]	D. M. Duc, T. V. Hung and N. T. Khai, Critical points of non-$C^2$ functionals, Topological Methods in Nonlinear Analysis, 29 (2007), 35-68.
[21]	P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier, Orientability of fredholm families and topological degree for orientable nonlinear fredholm mappings, J. Funct. Anal., 124 (1994), 1-39. doi: 10.1006/jfan.1994.1096.
[22]	N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, A.I.H.P. Analyse Non linéaire, 6 (1989), 321-330.
[23]	D. Gromoll and W. Meyer, On differentiable functions with isolated critical points, Topology, 8 (1969), 361-369.
[24]	H. Hofer, The topological degree at a critical point of mountain pass type, in "Nonlinear functional analysis and its applications, Part 1" (Berkeley, Calif., 1983), Proc. Symp. Pure Math., 45, Part 1, Providence, RI, (1986), 501-509.
[25]	D. Husemoller, "Fibre Bundle," Springer-Verlag, 1975.
[26]	A. Ioffe and E. Schwartzman, Parametric Morse lemmas for $C^{1,1}$-functions, 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.
[27]	M. Jiang, A generalization of Morse lemma and its applications, Nonlinear Analysis, 36 (1999), 943-960. doi: 10.1016/S0362-546X(97)00701-3.
[28]	E. Kreyszig, "Introduction Functional Analysis with Applications," John wiley & Sons. Ins. 1978.
[29]	S. Lang, "Differential Manifolds," $2^{nd}$ edition, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4684-0265-0.
[30]	C. Li, S. -J. Li and J. Liu, Splitting theorem, Poincare-Hopf theorem and jumping nonlinear problems, J. Funct. Anal., 221 (2005), 439-455. doi: 10.1016/j.jfa.2004.09.010.
[31]	C. Li, S.-J. Li, Z. Liu and J. Pan, On the Fucík spectrum, J. Differential Equations, 244 (2008), 2498-2528. doi: 10.1016/j.jde.2008.02.021.
[32]	G. Lu, Corrigendum to "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems" [J. Funct. Anal. 256(9)(2009)2967-3034], J. Funct. Anal., 261 (2011), 542-589. doi: 10.1016/j.jfa.2009.01.001.
[33]	G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces, preprint, arXiv:1102.2062.
[34]	G. Lu, Some critical point theorems and applications, preprint, arXiv:1102.2136.
[35]	G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds, preprint, arXiv:1212.2078.
[36]	Jean Mawhin and Michel Willem, On the generalized Morse Lemma, Bull. Soc. Math., Belgique (B), 37 (1985), 23-29.
[37]	Jean Mawhin and Michel Willem, "Critical Point Theory and Hamiltonian Systems," Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.
[38]	A. A. Moura and F. M. de Souza, A Morse lemma for degenerate critical points with low differentiability, Abstract and Applied Analysis, 5 (2000), 113-118. doi: 10.1155/S1085337500000245.
[39]	J. Pejsachowicz and P. R. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$, J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939.
[40]	K. Perera, R. P. Agarwal and Donal O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators," Mathematical Surveys and Monographs 161, American Mathematical Society, Providence Rhode Island 2010.
[41]	M. Schechter, "Principles of Functional Analysis," Academic Press, New York-London, 1971.
[42]	W. Schirotzek, "Nonsmooth Analysis," Springer, Berlin, 2007. doi: 10.1007/978-3-540-71333-3.
[43]	I. V. Skrypnik, "Nonlinear Elliptic Equations of a Higher Order," [in Russian], Naukova Dumka, Kiev 1973.
[44]	I. V. Skrypnik, "Nonlinear Elliptic Boundary Value Problems," Teubner, Leipzig, 1986.
[45]	A. Tromba, A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem, Math. Ann., 263 (1983), 303-312. doi: 10.1007/BF01457133.
[46]	S. A. Vakhrameev, Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems, J. Sov. Math., 67 (1993), 2713-2811. doi: 10.1007/BF01455151.

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