Palais-Smale condition for multi-time actions that produce Poisson-gradient PDEs

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1 Palais-Smale condition for multi-time actions that produce Poisson-gradient PDEs Iulian Duca Abstract. We study the Palais-Smale (P S) c -condition in the case of the multi-times actions ϕ that produces Poisson-gradient PDEs, where c is minimum value of ϕ on a Hilbert space. In Section we will present the well known bound between the (P S) c - condition and the existance of actional functional s extremas and we establish some conditions in which a function has a minimum in a reflexiv Banach space (heorem ). In Section we will prove the existance of the multiple periodical extremals of an action that produce Poisson-gradient systems (heorems 3 and 4) and we will show that the (P S) c -condition is satisfied, for the same action (heorem 5). M.S.C. 000: 53C, 49K0, 49S05. Key words: Palais-Smale (P S) c -condition, multi-time action, periodical solutions, PDEs Poisson-gradient. (P S) c -condition and minimum value for action in Banach spaces Let ϕ : X R be a differentiable function, where X is a Banach space. For c R, we say that ϕ satisfies the (P S) c -condition if the existence of sequence (u k ) in X such that ϕ (u k ) c, ϕ (u k ) 0, k, implies that c is a critical value of ϕ. We denote by W, the Sobolev space of the functions u L [, R n ], having weak derivatives u t L [, R n ], where [ 0, ]... [0, p ] R p and (,..., p). he weak derivatives are defined using the space C of all indefinitely differentiable multiple -periodic function from R p into R n. We consider the Hilbert space H, associated to the space W. he euclidean structure on H is given by the scalar product u, v (δ ij u i (t) v j (t) + δ ij δ αβ ui vj (t) tα t β (t) ) dt... dt p Proceedings of he 4-th International Colloquium Mathematics in Engineering and Numerical Physics October 6-8, 006, Bucharest, Romania, pp c Balkan Society of Geometers, Geometry Balkan Press 007.

2 64 Iulian Duca and the associated Euclidean norm. hese are induced by the scalar product (Riemannian metric) ( ) δij 0 G 0 δ αβ δ ij on R n+np (see the jet space J (, R n )). We denote by H {u H } u (t) dt... dt p 0, the Hilbert space of all functions from H which have mean zero. Proposition. [] Let ϕ : X R be a function bounded from below and continuously differentiable on a Banach space X. hen, for each minimizing sequence (u k ) of ϕ, there it exists a minimizing sequence (v k ) of ϕ such that ϕ (v k ) ϕ (u k ), u k v k 0, ϕ (u k ) 0, k. heorem. Let ϕ : X R be a continuous, convex and bounded from below function. If X is a Banach reflexive space and ϕ has a minimizing bounded sequence, then ϕ has a minimum value in X. Critical point actions (P S) c -condition for actions that produces Poisson-gradient systems We will prove the existence of a critical point and of the (P S) c -condition for the action ϕ (u) [ ] u t + F (t, u (t)) dt... dt p, on H. heorem 3. We consider F : R n R a function which satisfies the conditions:. F (t, x) is measurable in t for any x R n and with continuous derivatives in x for any t,. It exists a C (R +, R + ) with the derivative a bounded from above and b C (, R + ) such that for any x R n and any t, 3. x F (t, x) < h (t) for F (t, x) a ( x ) b (t), x F (t, x) a ( x ) b (t), h L (, R n ) ; t, x R n, 4. F (t, x) dt... dt p when x. We consider [ ] ϕ (u) u t + F (t, u (t)) dt... dt p.

3 Palais-Smale condition 65 If exists a sequence (u k ) in H, such that ϕ (u k) c, ϕ (u k ) 0 when k, then the sequence (u k ) is bounded from below on H. Proof. We write u k u k + ũ k, where u k... p u k (t) dt... dt p and we use the fact that it exists k 0 such that (.) ϕ (u k ), v v, k > k 0 because ϕ (u k ) 0. From the inequality () we have ϕ (u k ), ũ k ( xf (t, u k (t)), ũ k (t)) dt... dt p + ( ũk t (t), ũ ) k t (t) dt... dt p (.) ũ k By using [, heorem ] we find (.3) ũ k ũ k (t) dt... dt p + (( maxi { i }) 4π + ) ũ k t dt... dt p ũ k t dt... dt p By the inequality F (t, x) < h (t) from the hypothesis and Cauchy-Schwartz we obtain ( ) (h (t)) dt... dt p ( ũk ) L ( ) (h (t)) dt... dt p ( ) ũ k dt... dt p ( ) F (t, u k ) dt... dt p ( ) ũ k dt... dt p (.4) ( F (t, u k (t)), ũ k (t)) (t, u k ) dt... dt p. Using the inequalities (), (3) and (4) we have ( ) (h (t)) dt... dt p ) (( { ( ũ }) ) k maxi i L + 4π + ũ k ũ k, so

4 66 Iulian Duca h L ũ k + From here, it results that (( maxi { i }) 4π + (.5) ũ k C, ) ũ k ũ k. for k > k 0. Because ϕ (u k ) c, ϕ (u k ) bounded and we get the result ϕ (u k ) u k t u k t (t) u k t (t) + dt... dt p + F (t, u k (t)) dt... dt p dt... dt p + F (t, u k (t)) dt... dt p + [F (t, u k (t)) F (t, u k )] dt... dt p 0 dt... dt p + F (t, u k ) dt... dt p F (t, u k + sũ (t)) dsdt... dt p C. Using the relation (5) and the propertie 3 from hypothesis, we obtain (.6) u k C 4, k N, because we know that F (t, x) dt... dt p when x. From the relations (5) and (6) (u k ) is bounded in H. heorem 4. Some hypothesis as in heorem 3. If ϕ (u) F (t, u (t)) dt... dt p is weakly lower semi-continuous, then it exist u H such that ϕ (u) min ϕ (v). v H Proof. According to the heorem 3, the action ϕ is bounded from below on H. We will note by c inf ϕ (v). he action ϕ (u) δ αβ u δ i u j ij t dt... dt p is α t β v H weakly lower semi-continuous because is convex action on reflexiv Banach space H. Consequently, ϕ (u) ϕ (u) + ϕ (u) is weakly lower semi-continuous. Let (v k ) be a minimizing sequence for ϕ in H. According to the Proposition there it exists a minimizing sequence (u k ) such that ϕ (u k ) ϕ (v k ), u k v k 0, ϕ (u k ) 0, k. From the heorem 3 the sequence (u k ) is bounded on H and by the heorem it exist a minimum value ϕ (u) on H.

5 Palais-Smale condition 67 inf v H heorem 5. If the action ϕ verifies the properties from the heorem 4 and c ϕ (v) then ϕ satisfies the (PS) c-condition. Proof. We consider the sequence u k in H which satisfies the properties ϕ (u k) c, ϕ (u k ) 0 when k. his means that u k is a minimizing sequence for ϕ. From the heorem 4 it exists u H such that ϕ (u) c. According to [7, heorem 3] ϕ is continuously differentiable and from here it results that ϕ (u) 0, meaning that c is a critical value for ϕ. As consequence, ϕ satisfies the (P S) c-condition. References [] I. Duca, C. Udrişte, Some inequalities satisfied by periodical solutions of multitime Hamilton equations, he 5-th Conference of Balkan Society of Geometer, August 9 Sept., 005, Mangalia, Romania. [] J. Mawhin, M. Willem, Critical Point heory and Hamiltonian Systems, Springer-Verlag, 989. [3] C. Udrişte, From integral manifolds and metrics to potential maps, Conference Michigan State University, April 3-7, 00; Eleventh Midwest Geometry Conference, Wichita State University, April 7-9, 00, Atti del Academia Peloritana dei Pericolanti, Clase di Scienze Fis. Mat. e Nat., 8-8, A 0006 ( ), -4. [4] C. Udrişte, Nonclassical Lagrangian dynamics and potential maps, he Conference in Mathematics in Honour of Professor Radu Roşca on the Occasion of his Ninetieth Birthday, Katholieke University Brussel, Katholieke University Leuven, Belgium, Dec.-6, 999; [5] C. Udrişte, Solutions of DEs and PDEs as potential maps using first order Lagrangians, Centenial Vrânceanu, Romanian Academy, University of Bucharest, June 30-July 4, (000); Balkan Journal of Geometry and Its Applications 6, (00), [6] C. Udrişte, I. Duca, Periodical solutions of multi-time Hamilton equations, Analele Universitaţii Bucureşti, 55, (005), [7] C. Udrişte, I. Duca, Poisson-gradient dynamical systems with bounded nonlinearity, Communication at he 8-th International Conference of ensor Society, Varna aug. -6, 005. [8] C. Udrişte, M. Ferrara, D. Opriş, Economic Geometric Dynamics, Geometry Balkan Press, Bucharest, 004. [9] C. Udrişte, M. Neagu, From PDE systems and metrics to generalized field theories, [0] C. Udrişte, M. Postolache, Atlas of Magnetic Geometric Dynamics, Geometry Balkan Press, Bucharest, 00. Author s address: Iulian Duca University Politehnica of Bucharest, Faculty of Applied Sciences, Department Mathematics II, Splaiul Independentei 33, RO-06004, Bucharest, Romania. duca iulian@yahoo.fr