aa r X i v : . [ m a t h . F A ] N ov ON EKELAND’S VARIATIONAL PRINCIPLE
MARCO SQUASSINA
Abstract.
For proper lower semi-continuous functionals bounded below which do notincrease upon polarization, an improved version of Ekeland’s variational principle canbe formulated in Banach spaces, which provides almost symmetric points. Introduction
In the study of nonconvex minimization problems, the idea of looking for a specialminimizing sequence with good properties in order to guarantee the convergence towardsa minimizer goes back to the work of Hilbert and Lebesgue [7, 8]. In this direction one ofthe main contributions of the last decades in the calculus of variations was surely providedby Ekeland’s variational principle for lower semi-continuous functionals on metric spaces,discovered in 1972 [4, 5]. Since then, it has found a multitude of applications in differ-ent fields of nonlinear analysis and turned out to be fruitful in simplifying and unifyingthe proofs of already known results. We refer the reader to the survey [6] and to themonograph [2] for a discussion on a broad range of applications, including optimization,control and geometry of Banach spaces. The aim of the present note is that of showingthat within the abstract symmetrization framework proposed by Van Schaftingen in thenice paper [9], under the assumption that the functional does not increase by polariza-tion, the conclusion of the principle can be enriched with the useful information that theexisting almost critical point is almost symmetric as well. Furthermore, in the context ofthe critical point theory for nonsmooth functionals originally developed in [3], the resultallows to detect a Palais-Smale sequence ( u h ) in the sense of weak slope, which becomesmore and more symmetric, as h → ∞ , yielding a symmetric minimum point, providedthat some compactness is available. The additional feature often gives rise to a compact-ifying effect through suitable compact embeddings of spaces of symmetric functions. Forminimax critical values of C functionals, a similar result has been obtained in [9] underthe assumption that the functional enjoys a mountain pass geometry. Recently the wholemachinery has been extended by the author in [10] to a class of lower semi-continuousfunctionals in the framework of [3]. Deriving a symmetric version of Ekeland’s variationalprinciple in Banach spaces is easier than obtaining the results of [9, 10], since handling aminimization sequence is of course simpler than managing a minimaxing sequence. On theother hand, due to the great impact of Ekeland’s principle in the mathematical literatureover the last three decades, the author believes that highlighting the precise statementscould reveal useful for various applications. Let us now come to the formulation of theresults. Let X and V be two Banach spaces and S ⊆ X . We shall consider two maps Mathematics Subject Classification.
Key words and phrases.
Ekeland’s principle, nonconvex minimization, weak and strong slope.Research supported by PRIN:
Metodi Variazionali e Topologici nello Studio di Fenomeni non Lineari . ∗ : S → S , u u ∗ , the symmetrization map, and h : S × H ∗ → S , ( u, H ) u H , thepolarization map, H ∗ being a path-connected topological space. We assume, according to[9, Section 2.4], that the following hold:(1) X is continuously embedded in V ;(2) h is a continuous mapping;(3) for each u ∈ S and H ∈ H ∗ it holds ( u ∗ ) H = ( u H ) ∗ = u ∗ and u HH = u H ;(4) there exists ( H m ) ⊂ H ∗ such that, for u ∈ S , u H ··· H m converges to u ∗ in V ;(5) for every u, v ∈ S and H ∈ H ∗ it holds k u H − v H k V ≤ k u − v k V .Moreover, the mapping ∗ : S → V can be extended to ∗ : X → V by setting u ∗ := (Θ( u )) ∗ for every u ∈ X , where Θ : ( X, k · k V ) → ( S, k · k V ) is a Lipschitz function, of Lipschitzconstant C Θ >
0, such that Θ | S = Id | S . We refer to [9, Section 2.4] for some examples ofconcrete situations suitable in applications to partial differential equations.We recall [9, Corollary 3.1] a useful result on approximation of symmetrizations. Proposition 1.1.
For all ρ > there exists a continuous mapping T ρ : S → S such that T ρ u is built via iterated polarizations and k T ρ u − u ∗ k V < ρ , for all u ∈ S . In the above framework, here is the main result.
Theorem 1.2.
Assume that f : X → R ∪ { + ∞} is a proper and lower semi-continuousfunctional bounded from below such that (1.1) f ( u H ) ≤ f ( u ) for all u ∈ S and H ∈ H ∗ . Let u ∈ S , ρ > and σ > with f ( u ) < inf X f + ρσ. Then there exists v ∈ X such that (a) k v − v ∗ k V ≤ Cρ ; (b) k v − u k ≤ ρ + k T ρ u − u k ;(c) f ( v ) ≤ f ( u ) ; (d) f ( w ) ≥ f ( v ) − σ k w − v k , for all w ∈ X, for some positive constant C depending only upon V, X and Θ . Denoting by | df | ( u ) the weak slope [3] of f at u (it is | df | ( u ) = k df ( u ) k X ′ if f is C ), wesay that ( u j ) ⊂ X is a symmetric Palais-Smale sequence at level c ∈ R (( SP S ) c -sequence)if | df | ( u j ) → f ( u j ) → c and, in addition, k u j − u ∗ j k V → j → ∞ . We say that f satisfies the symmetric Palais-Smale condition at level c (( SP S ) c in short), if any ( SP S ) c sequence has a subsequence converging in X . In this context, we have the following Corollary 1.3.
Assume that f : X → R ∪ { + ∞} is a proper and lower semi-continuousfunctional bounded from below which satisfies (1.1) . Moreover, assume that for all u ∈ X there exists ξ ∈ S with f ( ξ ) ≤ f ( u ) . Then, for any ε > , there exists v ∈ X such that (a) k v − v ∗ k V ≤ Cε ; (b) f ( v ) < M + ε , M := inf f ; (c) | df | ( v ) ≤ ε ,for some C > . In particular, f has a ( SP S ) M -sequence. If f satisfies ( SP S ) M then f admits a critical point (for the weak slope) z ∈ X with f ( z ) = M and z = z ∗ . N EKELAND’S VARIATIONAL PRINCIPLE 3
This statement sounds particularly useful for applications to PDEs, via the additionalcontrol (a) that often gives rise to compactifying effects (see, for instance, the arguments of[9, pages 479 and 480]) via suitable compact embeddings of spaces of symmetric functions(we refer, for instance, to [11, Section I.1.5]). We note that the assumption that, for all u ∈ X , there exists an element ξ ∈ S such that f ( ξ ) ≤ f ( u ) is satisfied in many typicalconcrete situations, like when X is a Sobolev space, S is the cone of its positive functionsand the functional satisfies f ( | u | ) ≤ f ( u ), for all u ∈ X .Finally, let B ∗H ∗ the set of ϕ ∈ X ∗ such that k ϕ k ≤ h ϕ, u i ≤ h ϕ, u H i for u ∈ S and H ∈ H ∗ and for any u ∈ X there is ξ ∈ S with h ϕ, u i ≤ h ϕ, ξ i , being X ∗ the dual of X .In the spirit of [5, Corollary 2.4], Corollary 1.3 yields the following density result. Corollary 1.4.
Assume that f : X → R is a Gˆateaux differentiable function satisfyingthe assumptions of Corollary 1.3. Moreover, let α > and β ∈ R be such that f ( v ) ≥ α k v k + β, for all v ∈ X. Then, if S := { v ∈ X : k v − v ∗ k V ≤ } , the set df ( S ) ⊂ X ∗ is dense in α B ∗H ∗ . Proofs
Proof of Theorem 1.2.
Let u ∈ S , ρ > σ > f ( u ) < inf f + ρσ .If T ρ : S → S is the continuous mapping of Proposition 1.1, we set ˜ u := T ρ u ∈ S . Then,by construction we have k ˜ u − u ∗ k V < ρ and, in light of (1.1) and the property that ˜ u isbuilt from u through iterated polarizations, we obtain f (˜ u ) < inf X f + ρσ. By Ekeland’s variational principle (cf. [4, 5, 6]), there exists an element v ∈ X such that f ( v ) ≤ f (˜ u ) , k v − ˜ u k ≤ ρ, f ( w ) ≥ f ( v ) − σ k w − v k , for all w ∈ X. Hence (d) holds and, since f ( v ) ≤ f (˜ u ) ≤ f ( u ), conclusion (c) follows as well. In theabstract symmetrization framework, it is readily seen that k u ∗ − v ∗ k V ≤ C Θ k u − v k V forall u, v ∈ X . Then, if K >
X ֒ → V , it follows k v − v ∗ k V ≤ k v − ˜ u k V + k ˜ u − u ∗ k V + k u ∗ − v ∗ k V ≤ K ( C Θ + 1) k v − ˜ u k + k ˜ u − u ∗ k V ≤ ( K ( C Θ + 1) + 1) ρ, where we used the fact that u ∗ = ˜ u ∗ , in light of (3) of the abstract framework and, again,by the way ˜ u is built from u . Then, also conclusion (a) holds true. Finally, we have(2.1) k v − u k ≤ k v − ˜ u k + k ˜ u − u k ≤ ρ + k T ρ u − u k , yielding (b). This concludes the proof of the theorem. (cid:3) Proof of Corollary 1.3.
Given ε >
0, let u ∈ X be such that f ( u ) < M + ε . Byassumption, we can find an element ˆ u ∈ S such that(2.2) f (ˆ u ) < M + ε . Then, we are allowed to apply Theorem 1.2 with σ = ρ = ε and get an element v ∈ X such that k v − v ∗ k V ≤ Cε for some positive constant C depending only upon V, X and Θ,
MARCO SQUASSINA f ( v ) ≤ f (ˆ u ) and f ( w ) ≥ f ( v ) − ε k w − v k , for all w ∈ X . Whence, by taking into accountinequality (2.2), conclusions (a) and (b) of the corollary hold true. Moreover, sincelim sup w → v f ( v ) − f ( w ) k v − w k ≤ ε, by the definition of strong slope |∇ f | ( u ) [3] it follows |∇ f | ( u ) ≤ ε . Hence, since the weakslope satisfies | df | ( u ) ≤ |∇ f | ( u ) [3], assertion (c) immediately follows. Choosing now asequence ( ε j ) ⊂ R + with ε j → j → ∞ , by definition one finds a ( SP S ) M -sequence( v j ) ⊂ X . If f satisfies ( SP S ) M , there exists a subsequence, that we still denote by ( v j ),which converges to some z in X . Hence, via lower semi-continuity, we get M ≤ f ( z ) ≤ lim inf j →∞ f ( v j ) = M . Since | df | ( v j ) → f ( v j ) → f ( z ) = M as j → ∞ , by means of [3, Proposition 2.6], itfollows that | df | ( z ) ≤ lim inf j | df | ( v j ) = 0. Notice that, since k v j − v ∗ j k V → j → ∞ ,letting j → ∞ into the inequality k z − z ∗ k V ≤ k z − v j k V + k v j − v ∗ j k V + k v ∗ j − z ∗ k V ≤ ( C Θ + 1) K k v j − z k + k v j − v ∗ j k V , yields z = z ∗ , as desired. This concludes the proof of the corollary. (cid:3) References [1] F.J. Almgren, E.H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer.Math. Soc. 2 (1989), 683-773.[2] J.-P. Aubin, I. Ekeland, Applied nonlinear analysis. Pure and Applied Mathematics, Wiley & Sons,Inc., New York, 1984.[3] M. Degiovanni, M. Marzocchi, A critical point theory for nonsmooth functionals, Ann. Mat. PuraAppl. 167 (1994), 73-100.[4] I. Ekeland, Sur les probl´emes variationnels, C. R. Acad. Sci. Paris S´er. 275 (1972), 1057-1059.[5] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.[6] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474.[7] D. Hilbert, Uber das Dirichletsche Prinzip, Jber. Deut. Math. Ver. 8 (1900), 184-188.[8] H. Lebesgue, Sur le probl´ems de Dirichlet, Rend. Circ. Mat. Palermo 24 (1907), 371-402.[9] J. Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math. 7 (2005),463-481.[10] M. Squassina, Radial symmetry of minimax critical points for nonsmooth functionals, Commun.Contemp. Math., to appear.[11] M. Willem, Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, Birkh¨auser, Boston, 1996.
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