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Jean Mawhin’s contributions to critical point theory
Boundary Value Problemsvolume 2013, Article number: 115 (2013)
The aim of this article is to describe some fundamental contributions of Jean Mawhin to critical point theory and its applications to boundary value problems.
Dedicated to Jean Mawhin on the occasion of his seventieth birthday with friendship.
The first paper by Jean Mawhin on critical point theory  was published in 1982 and was devoted to periodic solutions of a forced pendulum equation. One of the most recent papers in 2012  concerns periodic solutions of difference systems with ϕ-Laplacian. It is impossible to describe all the contributions. We have selected 17 articles, 2 books and some fundamental topics:
– the forced pendulum equation,
– convex perturbations of indefinite quadratic functionals,
– construction of almost critical points,
– converse to the Lagrange-Dirichlet theorem, and
– Neumann problems for the ϕ-Laplacian.
1 From the classical to the relativistic pendulum
The forced pendulum equation is an important field of investigations of Jean Mawhin. We describe only some contributions (by variational methods) to the conservative forced pendulum, and we refer to the exhaustive survey  for other results.
Consider the classical second-order problem:
where f is 2π-periodic. The solutions of (1) are the critical points of the action functional
where . The space is the space of absolutely continuous functions u which are T-periodic and such that . Assuming that
it is not difficult to prove that Ψ achieves its infimum on and, consequently, that (1) is solvable. This result, due essentially to Hamel in 1922, was rediscovered by Willem in 1981 and by Dancer in 1982.
Some sixty years after the first one, a second periodic solution was discovered in  under assumption (2).
Since, by assumption (2),
a natural space of definition for Ψ is
The functional Ψ is bounded from below on X and, by a category argument, has at least two geometrically distinct critical points. A generalization to systems is contained in .
The argument in  was to use a refinement of the mountain pass theorem, observing that if v is a minimizer of Ψ, then, for all ,
Another proof, using a generalization of the Poincaré-Birkhoff theorem, was suggested by Franks . However, this proof is not complete . It seems that the variational proof is the only one until now. To find a proof using a fixed point theorem is an interesting challenge. Moreover, there is no exhaustive description of the set of h such that (1) is solvable assuming that f is 2π-periodic and (see  and ).
Let us recall the general notion of ϕ-Laplacian. Let (classical), or (bounded), or (singular) be an increasing homeomorphism such that . Canonical examples are, respectively, as follows:
The case of the p-Laplacian for the problem
was recently solved by Jean Mawhin in . The results are similar to the classical pendulum.
Consider now the forced relativistic pendulum and assume that
we define the action
Let us describe the recent results (2010) of Mawhin and Brezis on the relativistic pendulum . We sum up the simple and beautiful proof.
Theorem 1.1 Under assumptions (2) and (4), problem (3) has a solution which minimizes Ψ on C.
Lemma 1.2 The action Ψ has a minimizer on C.
Proof Let be such that . Since , we can assume that
It is clear that since . By going if necessary to a subsequence, we can assume that uniformly on ℝ and . It remains only to prove that
For any , we have that
We conclude by letting . □
Let us recall the notion of critical point in the sense of Szulkin .
Definition 1.3 Let X be a Banach space and let , where and is convex, proper (i.e., ) and lower semi-continuous (l.s.c. in short). A point is a critical point of I if and, for all ,
The easy proof of the next lemma is given in .
Lemma 1.4 Each local minimum of is a critical point of I.
We conclude the proof by using an argument due to Bereanu, Jebelean and Mawhin .
Proof of Theorem 1.1 Let u be a minimizer of Ψ on C. We have only to prove that in order to verify the Euler equation.
Let us define on
By an explicit computation, the problem
has exactly one solution w and . Since is strictly convex, w is the only solution of
But, by Lemma 1.4, we have that
We conclude that and
The case of Lagrangian systems of relativistic oscillators was recently treated by Mawhin and Brezis in .
An open problem from  is the extension in higher dimensions, for example,
2 Convex perturbations of indefinite quadratic functionals
in a closed subspace V of , where Ω is a bounded domain of . The linear operator is self-adjoint and the nonlinear potential is convex in its second variable.
It is always assumed that F is dominated at infinity by , with the first positive eigenvalue of L, and that there exists such that
Let us denote by K the inverse of
The dual action is defined on by
where is the Fenchel transform of :
Let . The perturbed dual action is defined on by
where is the Fenchel transform of .
It is assumed that K is the sum of a compact and of a positive definite operator. Because of the non-resonance condition with respect to , for small, is coercive on and has a minimizer . It suffices then to use the interaction between F and the kernel of L given in (6) to prove a posteriori estimates on . Passing to the limit as , we obtain a minimizer v of Ψ and, by duality, a solution u of (5).
Let Ω be a smooth bounded domain of and let be a Caratheodory function such that
We denote by the first positive eigenvalue of
We assume that
uniformly for, i.e., .
We consider the problem
Theorem 2.1 
(a) is nondecreasing for almost all ,
(b) on a subset of Ω of a positive measure.
Then problem (7) is solvable if and only if there exists such that
A similar result for the Dirichlet problem
is contained in .
The general results of  are applied to Dirichlet problems, Neumann problems and to periodic solutions of Hamiltonian systems and hyperbolic semilinear equations. In the latter case, the dimension of the kernel of L is infinite. See the survey  by Brezis.
It is important to note that the non-resonance assumptions are related to the potential , not to the gradient . In particular, the Palais-Smale condition is not necessarily satisfied.
General non-resonance conditions are used in  in order to prove the existence and uniqueness for semilinear equations in a Hilbert space by variational or iterative methods. Applications are given to semilinear wave equations.
3 Two books
We describe some main features of two books by Jean Mawhin devoted to critical point theory.
The book Problèmes de Dirichlet variationnels non linéaires (1986) is a nice introduction to critical point theory. The main tools,
– dual least action principle,
– minimax methods, and
– Morse theory,
are applied to the simple model problem
A new methodology was used in the construction of Palais-Smale sequences.
Definition 3.1 Let X be a Banach space and let . A Palais-Smale sequence (at level c) is a sequence such that
The Palais-Smale condition (at level c) is satisfied if every Palais-Smale sequence (at level c) contains a convergent subsequence.
Let us also mention the recent survey  on the Palais-Smale condition.
As written in the introduction of , the usual minimax method
1. prove an a priori compactness condition, like the Palais-Smale condition,
2. prove a deformation lemma depending upon this condition, and
3. construct a critical value,
could be replaced by the following steps:
1. prove a quantitative deformation lemma,
2. construct a Palais-Smale sequence, and
3. verify a posteriori compactness conditions.
The book  contains the first application of this methodology, using the quantitative deformation lemma in . (See  for another approach using Ekeland’s variational principle in the case of the mountain pass theorem).
The book Critical Point Theory and Hamiltonian Systems (1989) is motivated by the problems
Among many other results, a new bifurcation theorem is given. Consider the equation
where and . If there is some such that the critical groups satisfy
then there exists a bifurcation point for (8).
4 Converse to the Lagrange-Dirichlet theorem
In 1971, Hagedorn proved that, for Lagrangian systems of class , the equilibrium is unstable if it corresponds to a strict local maximum of the potential energy. The proof, using the theory of geodesics on Finsler manifolds, was rather involved. A new proof is given by Hagedorn and Mawhin in .
The idea is to replace Jacobi’s principle of least action by a new variational principle due to van Groesen . Let be the kinetic energy and let be the potential energy. The functional
is minimized on the subset
of for some suitable c.
5 Neumann problems for the singular ϕ-Laplacian
In this section, we describe some recent works motivated by the Neumann problem
The function satisfies assumption (4). Let us define
Then Szulkin’s critical point theory  is applicable to , since K is a convex l.s.c. function and since J is a differentiable function. The strategy is to prove that a critical point of I in the sense of Definition 1.3 satisfies and hence is a solution of (9).
Assume, for example, that
Then (9) is solvable if
The first case corresponds to a ground state of I and the second case to a saddle point of I (see ). The case of mountain pass solutions is also treated. The generalization of those results to the non-radial case is a challenging open problem.
Mawhin J: Periodic oscillations of forced pendulum-like equations.Lecture Notes in Math. 964. In Ordinary and Partial Differential Equations. Springer, Berlin; 1982:458-476.
Mawhin J: Periodic solutions of second order nonlinear difference systems with ϕ -Laplacian: a variational approach. Nonlinear Anal. 2012, 75(12):4672-4687. 10.1016/j.na.2011.11.018
Mawhin J: Global results for the forced pendulum equation. In Handbook of Differential Equations. North Holland, Amsterdam; 2004:533-589.
Mawhin J, Willem M: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differ. Equ. 1984, 52(2):264-287. 10.1016/0022-0396(84)90180-3
Mawhin J: Forced second order conservative systems with periodic nonlinearity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1989, 6: 415-434.
Franks J: Generalizations of the Poincaré-Birkhoff theorem. Ann. Math. 1988, 128(2):139-151.
Franks J: Erratum to: ‘Generalizations of the Poincaré-Birkhoff theorem’. Ann. Math. 2006, 164(2):1097-1098.
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Mawhin J: Periodic solutions of the forced pendulum: classical vs relativistic. Matematiche 2010, 65(2):97-107.
Brézis H, Mawhin J: Periodic solutions of the forced relativistic pendulum. Differ. Integral Equ. 2010, 23(9-10):801-810.
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Bereanu C, Jebelean P, Mawhin J: Variational methods for nonlinear perturbations of singular φ -Laplacians. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 2011, 22: 89-111.
Brézis H, Mawhin J: Periodic solutions of Lagrangian systems of relativistic oscillators. Commun. Appl. Anal. 2011, 5(2-4):235-250.
Clarke F: Solution périodique des équations hamiltoniennes. C. R. Acad. Sci. Paris Sér. A-B 1978, 287(14):A951-A952.
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Mawhin J Séminaire de Mathématiques Supérieurs. In Problèmes de Dirichlet variationnels non linéaires. Presses de l’Université de Montréal, Montreal; 1987.
Mawhin J, Ward JR, Willem M: Variational methods and semilinear elliptic equations. Arch. Ration. Mech. Anal. 1986, 95(3):269-277.
Mawhin J, Willem M: Critical points of convex perturbations of some indefinite quadratic forms and semi-linear boundary value problems at resonance. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1986, 3(6):431-453.
Mawhin J, Ward JR Jr., Willem M: Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem. Proc. Am. Math. Soc. 1985, 93(4):667-674. 10.1090/S0002-9939-1985-0776200-X
Mawhin J, Willem M Aplied Mathematical Sciences 74. In Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.
Brézis H: Periodic solutions of nonlinear vibrating strings and duality principles. Bull. Am. Math. Soc. 1983, 8(3):409-426. 10.1090/S0273-0979-1983-15105-4
Fonda A, Mawhin J: Iterative and variational methods for the solvability of some semilinear equations in Hilbert spaces. J. Differ. Equ. 1992, 98(2):355-375. 10.1016/0022-0396(92)90097-7
Mawhin J, Willem M: Origin and evolution of the Palais-Smale condition in critical point theory. J. Fixed Point Theory Appl. 2010, 7(2):265-290. 10.1007/s11784-010-0019-7
Willem M Progress in Nonlinear Differential Equations and Their Applications 24. In Minimax Theorems. Birkhäuser Boston, Boston; 1996.
Willem M Trabalho de matemática. In Lectures on Critical Point Theory. Universidade de Brasília, Brasília; 1983.
Aubin JP, Ekeland I Pure and Applied Mathematics. In Applied Nonlinear Analysis. Wiley, New York; 1984.
Hagedorn P, Mawhin J: A simple variational approach to a converse of the Lagrange-Dirichlet theorem. Arch. Ration. Mech. Anal. 1992, 120(4):327-335. 10.1007/BF00380318
van Groesen E: Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy. J. Math. Anal. Appl. 1988, 132: 1-12. 10.1016/0022-247X(88)90039-X
Bereanu C, Jebelean P, Mawhin J: Multiple solutions for Neumann and periodic problems with singular ϕ -Laplacian. J. Funct. Anal. 2011, 261(11):3226-3246. 10.1016/j.jfa.2011.07.027
Mawhin J: Radial solutions of Neumann problem for periodic perturbations of the mean extrinsic curvature operator. Milan J. Math. 2011, 79: 95-112. 10.1007/s00032-011-0148-5
The author declares that he has no competing interests.