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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 910406, 18 pages
The Systems of Nonlinear Gradient Flows on Metric Spaces and Its Gamma-Convergence
Department of Mathematics, Fu Jen Catholic University, New Taipei City 24205, Taiwan
Received 2 July 2012; Accepted 2 September 2012
Academic Editor: Norimichi Hirano
Copyright © 2012 Mao-Sheng Chang and Bo-Cheng Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We first establish the explicit structure of nonlinear gradient flow systems on metric spaces and then develop Gamma-convergence of the systems of nonlinear gradient flows, which is a scheme meant to ensure that if a family of energy functionals of several variables depending on a parameter Gamma-converges, then the solutions to the associated systems of gradient flows converge as well. This scheme is a nonlinear system edition of the notion initiated by Sylvia Serfaty in 2011.
The theory of Gamma-convergence was introduced by De Giorgi in the 1970s. It has become both a standard criterion for the study of variational problems and one of the extremely important topics in the calculus of variations. Gradient flows which are defined on metric spaces were also noticed by De Giorgi, that is, the notion replacing “gradient flows in a differentiable structure” then that of “curve of maximal slope in metric space.” This notion was introduced in  then further developed in [2–5]. It turns out to be useful in many applications, in particular for defining gradient flows over the probability measure spaces equipped with the Wasserstein metric.
In 2004, Gamma-convergence of gradient flows on Hilbert spaces was introduced by Sandier and Serfaty in . This abstract method states that if a family of energy functionals -converges to a limiting functional , then, under suitable conditions, solutions of the gradient flow of converge to solutions of the gradient flow of . This scheme was used successfully for the dynamics of Ginzburg-Landau vortices (cf. ), the Cahn-Hilliard equation (cf. [7, 8]), and the Allen-Cahn equation (cf. ).
The notion of Gamma-convergence of gradient flows on metric spaces was initiated by Serfaty  in 2011. She presented and proved the following.
Proposition 1.1 (cf.  Gamma-convergence of gradient flows in the metric spaces setting). Let and be complete metric spaces. Let and be functionals defined on metric spaces and , respectively. Assume that there is a sense of convergence of to which can be general and with respect to the -liminf convergence of to : Let and be strong upper gradients of and , respectively. Assume in addition the following relations. (1)Lower bound on the metric derivatives: if , for then for all (2) Lower bound on the slopes: if , then Let be a -curve of maximal slope on for with respect to , such that , which is well prepared in the sense that Then is a -curve of maximal slope with respect to and
Obviously, this scheme (Proposition 1.1.) can be applied to single gradient flow problems only. To the best of our knowledge, nonlinear equations are more difficult than linear equations, the problem of system of differential equations is more complicated and more important than the problem of scalar equations, and the system of nonlinear gradient flows on metric spaces has not appeared elsewhere. This gives us a motivation for studying the systems of nonlinear gradient flows on metric spaces and then establishing its Gamma-convergence structure which can be applied to the problems of nonlinear gradient flow systems on metric spaces. This scheme can be regarded as a “nonlinear system” edition of the notion initiated by Sylvia Serfaty.
This paper is organized as follows. In Section 2, we introduce some necessary knowledge on gradient flows, basic definitions of absolutely continuous curve, metric derivative, strong upper gradient, and curve of maximal slope for functional. In Section 3, we establish the explicit structure of nonlinear gradient flow systems and Gamma-convergence of the systems of nonlinear gradient flows on metric spaces. Finally, we give two examples to illustrate a special case of our main results in Section 4.
2. Basic Definitions and Preliminaries
Let be a real Hilbert space with the corresponding norm and let be a functional defined on . We say that is Fréchet differentiable at if there exists (the space of all bounded linear functionals on ) such that where .
Note that if such an exists, then it is unique and we denote . In view of the Riesz representation theorem there exists a unique element such that Moreover . is called the differentiable of at (notice that it is a bounded linear functional on ). We denote and we call the gradient of at . Hence we have We say that is of class on (i.e., ) if the map is continuous on . If , then the directional derivative of at in direction exists and is given by Let be a differentiable curve in with . Then The evolution equation is called the gradient flow of on Hilbert space . Using the definitions of Hilbert spaces and Gradient flows, it is easy to show the following basic and useful lemma.
Lemma 2.1. Suppose that for each , is an inner product space with induced norm . Let one defines for each , in . Then one has the following. (i) We can see that is an inner product space with induced norm Thus, is a Hilbert space. (ii) Let be a functional defined on and let . An element (and denote ) is called the gradient of at on the Hilbert space if for every differentiable curve in satisfying , Let one denotes is called the gradient of at with respect to . Hence one has The evolution equation (the gradient flow of on ) can be expressed as the following system of gradient flows on Hilbert spaces:
Definition 2.2 (-absolutely continuous curve). Let be a complete metric space equipped with the distance . A mapping is called a -absolutely continuous curve or belongs to , , if there exists an function such that
Proposition 2.3 (cf. ). Let be a metric space and let . Then (i) if and only if there exists , , such that (ii) If , the metric derivative exists for a.e. , , and if satisfies (2.15), then a.e. on .
In the following, let us give a motivation for defining “gradient flow” on metric spaces. Here we completely follow the nice contents of Section 1.3 in . Note that every solution of the gradient flow can be characterized by the following the scalar equations: Using Young's inequality, (2.19) and (2.20) are equivalent to We can impose (2.18), (2.19), (2.20), and (2.21) as a system of differential inequalities in the couple by using the following strategies. (i) The function is an upper bound for the modulus of the gradient for every regular curve . (ii) Impose that the functional decreasing along as much as possible compatibly with (2.22), that is, (iii) Prescribe the dependence of on , or even in a single formula
Whereas (2.18), (2.19), and (2.20) make sense only in a Hilbert space framework, the formulas (2.21)~(2.25) are of purely metric nature and can be extended to more general metric space, provided we understand as the metric derivative of , . Of course, the concept of upper gradient provides only an upper estimate for the modulus of in the regular case, but it is enough to define steepest descent curves, that is, curves which realize the minimal selection of compatible with for a.e. .
Suppose that , and is Borel. Using (2.26), we have for any , Therefore we say that is a strong upper gradient for if for each , is Borel and (2.27) holds for all . Using the ideas (ii) and (iii) and Young's inequality, we say that a locally absolutely continuous function is a curve of maximal slope for with respect to its strong upper gradient if is a.e. equal to a nonincreasing map and for a.e. . Let us present the main definitions and three lemmas as follows.
Definition 2.4 (strong upper gradient). Suppose that is a complete metric space equipped with the distance for . Let for each with . We say that is a strong upper gradient for if for each , is Borel for , and
Definition 2.5 (a pair of Young's functions). Suppose that and are two differentiable functions. We say that is a pair of Young's functions if they satisfy(1) Young’s inequality: for all ,(2) Young’s equality: or ,
where and .
Definition 2.6 (curve of maximal slope). Let be a pair of Young's functions for each with . We say that is a -curve of maximal slope for the functional with respect to the strong upper gradient if is -a.e. equal to a nonincreasing map and for -a.e. .
Definition 2.7 (-liminf convergence). Let and be complete metric spaces for all and . Suppose that and are functionals defined on and , respectively. We say that -liminf converges to if then where , , and the sense of convergence can be general.
Lemma 2.8. Suppose that and for each . If and if is a continuous nondecreasing function on , then
Proof. (i) For each , , .
(ii) Since is non-decreasing on , and using (i), we have Therefore, and so (iii) Since is continuous on , and using (ii), we have
Lemma 2.9 (see [10, Theorem 5.11]). Let be nonnegative and measurable on . Then if and only if a.e. in .
Lemma 2.10. Let be a metric space for each . Let . Define the function by for each . Then(i) is a metric space, (ii) suppose that . The metric derivative can be expressed as for a.e. .
3. Main Results
In the following theorem we introduce the systems of explicit nonlinear gradient flows of energy functional with respect to the strong upper gradient on metric spaces and investigate an upper control for some form of velocity of solutions by its dissipation rate of the energy functional. Using this idea, we can see that if motion is driven by energy dissipation and if there are solutions that move without losing much energy, then they must move very slowly for each component solution.
Theorem 3.1. Let be a complete metric spaces equipped with distance for . Let be a functional defined on and let be a strong upper gradient for . Assume that is a continuous, strictly increasing and surjective function for each . Let and be defined by for each and . Suppose that and that is a -curve of maximal slope for the functional with respect to the strong upper gradient on . Then one has the following. (i) for a.e. . (ii) satisfies the system of explicit nonlinear gradient flows of with respect to the structure for a.e. . (iii) Assume that function is convex and strictly increasing on for each , then one has (a) for all , (b) for a.e. . (iv) If for , then (ii) can be expressed as which is the system of explicit linear gradient flows of with respect to the structure .
Proof. According to the definitions of and , is a pair of Young's functions. Recall we assume that is a -curve of maximal slope for with respect to on . For a.e. , we have
The last inequality holds due to the assumption of strong upper gradient for . Thus we easily obtain the following formula:
for a.e. . Using the Young's inequality and the Vanishing theorem (Lemma 2.9), we obtain
for a.e. and for . By Young's equality, we discover that the system of explicit gradient flows of with respect to the structure holds.
We now prove assertion (iii). By using assertions (i), (ii), and the assumption for , we see that for a.e. . Owing to the metric derivative which is the smallest admissible function satisfying and is convex and strictly increasing on , by using Jensen's inequality, we find This completes the proof of (a). By passing to the limit in (a), we deduce that (b) holds.
In our second main result, we study and establish the abstract structure of “Gamma-convergence of gradient flow systems on metric spaces” which is a nonlinear system edition of the notion (Proposition 1.1) and which can be applied to the problems involving a system of nonlinear gradient flows on metric spaces.
Theorem 3.2 (Gamma-convergence of systems of gradient flows on metric spaces). Let and be complete metric spaces for all and . Let and be functionals defined on spaces and , respectively. Suppose that the - convergence of to holds. Let and be strong upper gradients of and , respectively. Let be a pair of Young's functions having continuous, strictly increasing and surjective derivative for . Assume in addition the following relations. (1)Lower bound on the metric derivatives: if , for then for all (2)Lower bound on the strong upper gradients: If , then Let be a -curve of maximal slope on for with respect to such that , which is well-prepared in the sense that Then (i) is a -curve of maximal slope on for with respect to , (ii)(iii)(iv) where .
Proof. Owing to the fact that is a -curve of maximal slope for with respect to on , we recall that Theorem 3.1-(i) and (ii) yields
for a.e. ;
for a.e. and for .
Passing to the to (3.19) and applying Fatou's lemma, we deduce that The last inequality is achieved by the assumptions of (1) and (2) as well as Lemma 2.8.
Using the fact that each is a pair of Young's functions (hence Young's inequality holds), (3.21), and the strong upper gradient assumption of for , we can check that We recall that is well prepared, and using (3.22), we can deduce that
Using the -liminf convergence of to and (3.23), we obtain
Combining (3.24) with (3.22), we can now conclude that, for each , Using the Young's inequality and the Vanishing theorem (Lemma 2.9) again, we conclude that, for a.e. , for all . Moreover, by Young's equality, we have, for a.e. , for each .
Next, differentiating formulas (3.25) with respect to variable , we see that is a -curve of maximal slope for with respect to on . Using formula (3.19), (3.24), and (3.25) we can check that for all . Finally, we recall formulas (3.20) and (3.27), and using (3.28), we obtain and complete the proof of Theorem 3.2.
In this section, we present two examples to illustrate a special case of our main results.
Example 4.1. Let . If for each , then for , where . Hence In this case, Theorem 3.1 can be expressed as following. (i) One has (ii) The system of explicit nonlinear gradient flows of with respect to the structure is (iii) for ( is convex and strictly increasing on for each ). Then(a) for all ,(b) Under the hypotheses of Theorem 3.2, we have (i) that is, .(ii) that is, .
Example 4.2. Considering the case in Example 4.1, we have(i)(ii)The system of explicit linear gradient flows of with respect to the structure is (iii) for all . Moreover, we have (iv)(v)
Research by authors was supported by National Science Council of Taiwan under Grant no. NSC 100-2115-M-030-001.
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