On the Coercive Functions and Minimizers

Transcription

1 Advanced Studies in Theoretical Physics Vol. 11, 17, no. 1, HIKARI Ltd, On the Coercive Functions and Minimizers Carlos Alberto Abello Muñoz Universidad del Quindío Grupo de Modelación Matemática en Epidemiología Armenia, Colombia Pedro Pablo Cárdenas Alzate Department of Mathematics and GEDNOL Universidad Tecnológica de Pereira Pereira, Colombia José Rodrigo González Granada Department of Mathematics and GEDNOL Universidad Tecnológica de Pereira Pereira, Colombia Copyright c 17 Carlos Alberto Abello Muñoz, Pedro Pablo Cárdenas Alzate and José Rodrigo González Granada. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract N.G. Meyers significantly extended weak lower semicontinuity results for integral functionals depending on maps and their gradients available at that time. Special attention is paid to signed integrands and to applications to continuum mechanics of solids. In particular, we review existing results of the global minimizer of a coercive function and related statements about sequential weak continuity of minors []. Keywords: Coercive functions, sequentially lower semi-continuous, global minimizers.

2 71 Carlos Alberto Abello Muñoz et al. 1 Introduction We now know to prove that a critical point of a function f(x) is a global minimizer if the Hessian of f(x) is positive semidefinite on all of R n or a strict global minimizer if the Hessian is positive definite, but we need a way to establish global minimizers even if the Hessian is not necessarily positive semidefinite on all of R n. Previously [1], we defined a set D to be closed if its complement is open. This sets the stage for the following definitions. Definition 1. We then say that a set D R n is bounded if there exists a constant M > such that x < M for all x D. The set D is said to be compact if it is closed and bounded [,3]. Theorem 1. Let D be a compact subset of R n. If f(x) is a continuous function on D, then f(x) has a global maximizer and a global minimizer on D. We now describe functions for which global minimizers can be found even on sets that are not bounded or not closed. Definition. A continuous function f(x) that is defined on all of R n is coercive is lim f(x) = +. x Definition 3. The function F its weakly sequentially lower semi-continuous (w.s.l.s.c) at x M, if (x k ) k N M with x w k x (for x ) there holds [3,4] F (x ) F (x k). k That is, for any constant M > there exists a constant R M > such that f(x) > M whenever x > R M. Theorem. Let f(x) be a continuous function defined on all of R n. If f(x) is coercive, then f(x) has a global minimizer. Furthermore, if the first partial derivatives of f(x) exist on all of R n, then any global minimizers of f(x) can be found among the critical points of f(x) [5]. We can see that this theorem can be proved by using the fact that f(x) is coercive to find a compact subset of R n on which f(x) must have a global minimizer, by the preceding theorem. Therefore, to find the global minimizer of a coercive function, it is sufficient to find the critical points of f(x), and

3 On the coercive functions and minimizers 711 then evaluate f(x) at each of these points. The critical points for which f(x) assumes the smallest values are then the global minimizers [6]. Results on w.s.l.s.c functions Let φ(x) : H 1 T R be a function defined by φ(x) = x (t) + F (t, x(t)) dt. Here, F : [, T ] R N R is a function continuous such that: Let x F (t, x) be continuously differentiable for all t [, T ]; we denote ( F F (t, x) = (t, x),, F ) (t, x) ; x 1 x N There is a C([, [, [, [), b C[, T ] such that F (t, x) a( x )b(t), F a( x )b(t), (t, x) [, T ] R N. We will show that the function φ(x) is w.s.l.s.c. To show that φ is w.s.l.s.c, we must prove that if x n x then φ(x n) φ(x). Let s first write φ(x n ) in the form φ(x n ) = L(x n ) + T (x n ), then, we have that Note also that φ(x) we can write it as = 1 φ(x) = 1 φ(x n) L(x n) + T (x n). }{{}}{{} L(x) T (x) x (t) dt + x (t) + x(t) dt + = 1 x + F (t, x(t)) dt x(t) + F (t, x(t)) dt x(t) + F (t, x(t)) dt. Now, we know that if E is a Banach space, then if x n x = (x n) x. Let φ(x n) with x n x in HT 1. We will proceed in stages.

4 71 Carlos Alberto Abello Muñoz et al. 1. There is a subsequence {x nk } such that for all subsequence {x nk } in x n. φ(x n k ) = φ(x n),. By compactness of the injection i : H 1 ([, T ], R N ) C([, T ], R N ) we know that there is a subsequence {x nkl } which converges strongly in C([, T ]) since strong convergence implies weak convergence. 3. Let {x nkl } x in HT 1. As the injection i is compact, i.e., i of H1 in C([, T ]) is compact, so a sequence that tends weakly in H 1 converges for the norm of C([, T ]), therefore {x nkl } x in C([, T ]). 4. As {x nkl } x in C([, T ]), then x nkl (t) + F (t, x nkl (t)) dt 5. Now, as x n x = x x n then x n kl x. 6. Finally, steps i, iv and v can be concluded that φ(x n) = x (t) dt + F (t, x(t)) dt. φ(x n kl ) φ(x), and therefore φ(x) φ(x n ) for all sequence {x n } of E such that x n x. Now, if we want to prove that φ is differentiable in the sense of Fréchet, we can see that φ is well defined for all x H 1. Then, we know that φ(x) is given by φ(x) = 1 x (t) dt + F (t, x(t)) dt. Therefore, for the second integral, just use the maximum of F and the fact that H 1 C([, T ]), since [, T ] is bounded. Let us show that φ(x) is differentiable in the sense of Fréchet and that its defferential is given by φ ( x) = x x dx + f(x) x dx,

5 On the coercive functions and minimizers 713 for all x H 1 where F is the primitive of f which vanishes. Let s show the differentiability of the application x T (x) = F (t, x(t)) dt, then for all x, x H([, 1 T ]) T (x + x) T (x) = f(t, x(t)) x dt = ( 1 ( 1 ) (f(x + s x) ds f(t, x)) ds x dt. ) f(x + s x) x ds f(t, x) x dt d In fact, (F (x+s x)) = f(x+s x) x, so by the inequality of Cauchy-Schwarz ds we get that ( T (x + x) T (x) T 1/ f(t, x) x dt (f(x + s x) f(t, x))ds) dt) x L ([,T ]). By Cauchy-Schwarz we have ( (f(x + s x) f(t, x)) ds) 1 (f(x + s x) f(t, x)) ds, so we can get T (x + x) T (x) f(t, x) x dt f(x+s x) f(t, x) L ([,T ] [,1]) x L [,T ] f(x + s x) f(t, x) L ([,T ] [,1]) x H 1 [,T ]. Now, to conclude just show that for any sequence x n H([, 1 T ]), we have that which tends to in f(x + s x n ) f(x) L ([,T ] [,1]). Here we start by extracting a subsequence which is the upper limit of this sequence and which converges almost everywhere. So for this sequence we have f(x + s x n ) f(t, x) f L H1 ([[, T ] [, 1]]). It can be shown using the same type of arguments, that the application x φ (x) is continuous H 1 ([, T ]) in its dual (H 1 ([, T )).

6 714 Carlos Alberto Abello Muñoz et al. Now, we show that f is weakly coercive. For that, we must show that φ when x. Here, we can write F (t, x(t)) in the form F (t, x(t)) = F (t, x) + F (t, x(t)) F (t, x). Now, for x HT 1 we have by (a) that x = x+ x where x = (1/T ) x(t) dt, then φ(x) = = x (t) dt + x (t) dt + F (t, x) dt + F (t, x) + F (t, x(t)) F (t, x) dt 1 F (t, x(t)) F (t, x) dt x (t) = dt + F (t, x) dt + ( F (t, x + s x(t)), x(t)) ds dt x (t) ( ) dt + F (t, x) dt c(t) dt x, and by the Sobolev inequality we get x (t) dt + ( F (t, x) dt k therefore, we know that x and knowing that 3 Conclusion x (t) dt) 1/, ( x + x (t) dt) 1/ F (t, x) dt when x we can conclude that φ(x) when x. In this paper, we prove that φ is coercive using the fact that the functional 1 x(t) dt is convex and more is continuous, so knowing tha if φ is s.l.s.c and convexe, then φ is w.s.l.s.c. On the other hand, we prove that the functional F (t, x(t)) dt, so we know that if a sequence x n converges weakly towards x in HT 1, then x n converges uniformly to x in [, T ], i.e., x n x, then tends uniformly. This shows us that F (t, x n(t)) dt converges to F (t, x(t)) dt, so this functional is weakly continuous and therefore w.s.l.s.c. Acknowledgements. We would like to thank the referee for his valuable suggestions that improved the presentation of this paper and our gratitude to Grupo de Modelación Matemática en Epidemiología of the Universidad del Quindío and Department of Mathematics of the Universidad Tecnológica de Pereira (Colombia) and the group GEDNOL.

7 On the coercive functions and minimizers 715 References [1] A. Gopfert, Mathematische Optimierung in Allgemeinen Vektorraumen, Teubner Verlag, Leipzig, [] P. Cárdenas Alzate, Analyse Fonctionelle: Une Introduction, Des Notes de Classe, Université de Montréal, 15. [3] R.T. Rockafellar, Convex Analysis, Princeton University Press, New Jersey, [4] B. Barrios, E. Colorado, A. de Pablo, U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 5 (1), no. 11, [5] H. Brézis, Analyse Fonctionelle, Théorie at Applications, Masson, Paris, [6] Enrico Valdinoci, Raffaella Servadei, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 5 (13), Received: November 3, 17; Published: December 17, 17