THEre are many physical phenomena that can be described

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1 IAENG International Journal of Applied Mathematis, 8:, IJAM_8 3 On Young Funtionals Related to Certain Class of Rapidly Osillating Sequenes Andrzej Z. Grzybowski, Member, IAENG, Piotr Puhała Abstrat The paper is devoted to theory of lassial Young measures and assoiated with them Young funtionals i.e. ertain mappings that assign to any Carathodory funtion an integral with respet to related Young measure. The main result provides us with probabilisti haraterization of suh measures in the ase where assoiated sequene of fast osillating funtions has uniform representation - a notion introdued in this paper. In many interesting ases this new haraterization enables one to find an expliit formulae for the density funtions of the Young measures and thus also to ompute effetively the values of orresponding Young funtionals. In other ases, where the expliit formulae for the densities annot be reeived, the main result makes it possible to take advantage of Monte Carlo methods to evaluate the Young funtional s values. Index Terms Young measures, Young funtionals, fastosillating sequenes, periodi funtions, Monte Carlo. I. INTRODUCTION THEre are many physial phenomena that an be desribed and explained by analysis of the suitable funtionals orresponding to the respetive physial quantities. The natural way of analysis of suh funtionals an be performed with the use of variational methods, among of whih the minimization methods are one of the most powerful. Their usage an be justified as a onlusion from the variational laws of nature suh as, say, the least ation priniple. Therefore there is no surprise that variational methods have been proved useful in engineering. One may look at [1], [], [3], [5], [9], [1], or at books [11], [13] (and the referenes ited there) to mention just a few examples from vast literature on this subjet. In general, we minimize the energy funtional J of the form J (u) = H ( x, u(x), u(x) ) dx, where R d is a bounded Lipshitz domain; u: K R l is a funtion from a suitable Sobolev spae V (with K being a ompat set); and H: R d R ld R {+ } is a funtion satisfying ertain growth and regularity onditions. In the nonlinear elastiity theory we have d = l = 3, is the onsidered elasti body, u is the displaement of, H is density of the internal energy (it usually depends on some physial onstants). The funtional J is bounded below by nature so there exists a minimizing sequene for J, that is a sequene {u k } V suh that lim J (u k) = inf{j (v) : v V }. k Manusript reeived June 6, 17; revised June 15, 17. A.Z. Grzybowski and P. Puhała are with the Institue of Mathematis, Faulty of Mehanial Engineering and Computer Siene, Czestohowa University of Tehnology, Czestohowa, -1 Poland, andrzej.grzybowski@im.pz.pl, piotr.puhala@im.pz.pl However in general, for example when minimizing energy funtionals orresponding to the multi-well potentials, the integrand does not satisfy onvexity ondition guaranteeing ahieving by J its infimum. In that ase elements of the minimizing sequene are the funtions that osillate rapidly around the weak (or weak ) limit of that sequene. This situation is met in engineering for example when energy funtionals of ertain alloys are minimized. With the help of mirosope one an observe what is by engineers alled a mirostruture a struture somehow between the miroand marosoping sale. The following example shows the ruial role of rapid osillations in the arising of mirostruture, [13]. Consider a funtion sequene {sin kx}, k N, defined on an interval (, ). On one hand we have dx, as k, while on the other dx. sin(kx)dx sin (kx)dx This in partiular shows that even when the funtion ϕ is ontinuous and a funtion sequene u k satysfies lim u k (x)dx = u (x)dx, the equality lim ϕ(u k (x))dx = ϕ(u (x))dx does not have to be true in general. More generally we an say: the existene of the weak limit of a funtion sequene does not guarantee even the existene of the weak limit of the sequene whose elements are ompositions of the elements of the aforementioned sequene with fixed (even ontinuous) funtion. In the ontext of the variational methods a generalization of the above example is presented in [11], see also [7]. One of the ways of dealing with suh problems is the L. C. Young s idea: enlarging the spae of admissible funtions from Sobolev spaes to the spae of parametrized measures ν := {ν x } x,whih are now alled Young measures. Young measures theory has a long history. It starts with the seminal work [15] where the main onept was introdued with the aim to provide extended solutions for some non-onvex problems in variational alulus. After then Young developed his pioneering ideas in [16]. These generalized limits of osillating sequenes where alled by Young himself generalized urves in the one dimensional ase, and generalized surfaes in more general situation. Nowadays this onept has found appliations in the solution of various non-onvex optimization problems - that are at the ore of many different ontemporary engineering theories. They arise e.g. in optimal ontrol, nonlinear evolution equation, variational alulus, miromagneti phenomena in ferro-magneti materials as well as in mirostrutures theory in ontinuum mehanis. It turns out that even the simplest form of a Young measure, (Advane online publiation: 7 November 18)

2 IAENG International Journal of Applied Mathematis, 8:, IJAM_8 3 namely this where the measure does not depend on the parameter x, is important both in the theory and appliations. Suh Young measures are alled homogeneous ones. In ontemporary literature the Young measures are defined under different assumptions about underlying spaes and analyzed from different standpoints. This paper fouses on the lassial Young measures related to sequenes of rapidly osillating funtions. Presented here results are partly based on our onferene presentations [7], [8]. The struture of the paper is the following. In the next setion we introdue some preliminary definitions and results from the Young measure theory. In Setion III we define some lasses of fast-osillating sequenes and state new proposition that provide a diret link between the Young measure theory and the probability theory. Setion IV presents exemplary theoretial result that is a onlusion from the main proposition stated in Setion III. This onlusion allows one to find expliit forms of the density funtions of related lassial Young measures in various situations. Setion VI illustrates the usefulness of our new results in the problems of omputing Young funtionals values Finally we make some remarks about possible further extensions. II. PRELIMINARY DEFINITIONS AND RESULTS NOw we introdue basi notions of the Young measure theory from the point of view of nonlinear elastiity. Our presentation follows the approah taken in [13], where the reader is referred to for detailed information along with neessary notions from funtional analysis and further bibliography. Another book treating Young measures thoroughly in the ontext the optimization theory and variational alulus is [1]. Let be a nonempty, open and bounded subset of R d with smooth boundary. Denote by L () the Banah spae of essentially bounded funtions defined on with values in a ompat set K R l. Let {f n } be a sequene of funtions onverging to some funtion f weakly in L and denote by ϕ a ontinuous real valued funtion with domain R l. By the ontinuity of ϕ the sequene {ϕ(f n )} is uniformly bounded in L norm and Banah-Alaoglu theorem yields the existene of the (not relabeled) subsequene suh that ϕ(f n ) g weakly in L. However, in general g is not ϕ(f ), moreover, it is not even a funtion with domain in R l. To quote from [13]: The Young measure assoiated with {f n } furnishes the link among {f n }, f, g and ϕ. We now state the basi existene theorem for Young measures in its full generality. Theorem.1: (see Theorem. in [13]) Let R d be a measurable set and let z n : R l be measurable funtions suh that sup h( z n )dx <, n where h: [, ) [, ) is a ontinuous, nondereasing funtion suh that lim t h(t) =. There exist a subsequene, not relabeled, and a family of probability measures ν = {ν x } x (the assoiated Young measure) depending measurably on x with the property that whenever the sequene {H(x, z n (x))} is weakly onvergent in L 1 () for any Carathéodory funtion H(x, λ): R l R { }, the weak limit is the funtion H(x) = H(x, λ)dν x (λ). R l The family of probability measures ν = {ν x } x is alled the Young measure assoiated with the sequene {z n }. Let us reall that the Carathéodory funtion is a funtion that is measurable with respet to the first and ontinuous with respet to the seond variable. It often happens that the Young measure ν = {ν x } x does not depend on x. In this ase we denote it merely by ν; suh Young measure is alled homogeneous. One may also look at the Young measure as at objet assoiated with any measurable funtion defined on a nonempty, open, bounded subset of R d with values in a ompat subset K of R l. Suh a onlusion an be derived from the theorem in [1]. Due to this theorem it an be proved that the Young measure assoiated with a simple funtion is homogeneous and is the onvex ombination of Dira measures. These Dira measures are onentrated at the values of the simple funtion under onsideration while oeffiients of the onvex ombination are proportional to the Lebesgue measure of the sets on whih the respetive values are taken on by the funtion; see [1] for details and more general results onerning simple method of obtaining expliit form of Young measures assoiated with osillating funtions (similar, although mathematially more ompliated situation, is met in elastiity when the deformed body has a laminate struture; see e.g. Setion.6 in [11]). On the basis of this onept a more general haraterization of a Young measure assoiated with any Borel funtion was introdued in [6]. The main result stated there provides diret link between the Young measure onepts and the probability theory. Namely, the following theorem holds: Theorem.: Let f: R d K R l be a Borel funtion with Young measure ν. Then ν is the probability distribution of the random variable Y = f(u), where U has a uniform distribution on. Now, let us reall the notion of lassial Young measure assoiated with a sequene of osillating funtions {f k }, see e.g. [1]. Definition.1: The lassial Young measure generated by the sequene {f k } is a family of probability measures ν = {ν x } x satysfying the ondition: for any Carathéodory funtion H H(x, f k (x))dx k K H(x, y)dν x (y)dx (1) The appliation that assigns to any Carathéodory funtion H the integral given on the right-hand side of the above equation is alled Young funtional. Its values on H will be denoted here as YF(H) while the integrals on the left-hand side of 1 will be denoted as C(f k, H) Basially, the above definition presents the original understanding of the Young measure, as introdued in his work [16]. These measures are of our main onern in this paper. (Advane online publiation: 7 November 18)

3 IAENG International Journal of Applied Mathematis, 8:, IJAM_8 3 III. MAIN RESULT AND ITS THEORETICAL APPLICATIONS A. Rapidly osillating sequenes with uniform representation Let funtion f: [a, b) K R be a Borel funtion defined on the interval [a, b), b > a, and let f e : R K be the periodi extension of f (with the period equal to T = b a). Let be a given interval. A sequene {f k } of funtions f k : K, k = 1,,... defined by the formula f k (x) = f e (kx), x () will be alled a Rapidly Osillating Sequene with Uniform representation f, and denoted as ROSU(f). In suh a ase we will also say that f generates rapidly osillating sequene {f k }. Note, that the interval - the domain of elements of ROSU(f) - does not have to be the same as [a, b) i.e. the domain of f. Example 1 In this example we present illustrative plots of some elements of ROSU(f), where f(x) = (x ), x [, 13/) (3) Its purpose is not only to illustrate the behavior of ROSU (whih as a matter of fat is quite obvious) but also to illustrate the onept of lassial Young measure assoiated with the sequene {f k }. The plots of the funtion f given by (3) as well as of examplary elements of ROSU(f) with the domain = [, 5) are presented in Fig.1. Namely it shows the plots of f 1, f 5 and f 35. It is seen that the graphs of f k are getting denser when k tends to infinity. Unfortunately, a onventional weak* luster point of {f k } loses most of the information about the fast osillations in {f k } beause, in some sense, it takes into aount only the mean values of {f k } - as the integrals simply do. That is why we need a new onept of the limit and here the theory of Young measures appears to be very helpful. If ν x is the Young measure assoiated with {f k } then, roughly speaking, for any measurable set A K the intuitive meaning of ν x (A) is the probability that for an infinitesimally small neighborhood S of x and suffiiently large k s we an find f k (s) in A, when s hanges within S. In some sense it reflets the density of the values in K - it an be observed in the plot (d) (for f 35 ) where more frequent values reate darker straps. B. Charaterization of the Classial Young measures generated by the ROSU(f) It results diretly from the definition of ROSU(f) that its behaviour, as k tends to infinity, is exatly the same in every neighborhood of any x. In other words, its asymptoti behavior in an arbitrarily small interval I does not depend on where the interval is plaed within the domain. Consequently, it is obvious that the lassial Young measure generated by the ROSU(f) is the homogeneous Young measure (i.e. it does not depend on x ). Now, let U S denote a random variable uniformly distributed on S. Let us onsider two funtions g 1 : 1 K and g : K. We will say that the two funtions identially transform a uniform distribution if the distributions of the random variables g 1 (U 1 ) and g (U ), are the same. This fat will be denoted by g 1 g. Obviously the is the equivalene relation. Note that if the ROSU(f) is defined on the same interval as the generating funtion f, then any of its elements transform a uniform distribution identially as the funtion f, i.e. f k f for any k = 1,,... Now let us onsider the ase where the ROSU(f) is defined on interval that is different than the domain [a, b) of the generating funtion f. In suh a ase it an be seen that f k (U ) D f(u [a,b) ), where D denotes the onvergene in distribution, see []. In other words the distribution of f(u [a,b) ) is a vague limit of the distributions of f k (U ). Indeed, for k s large enough so the length of interval is greater than (b a)/k we have for any measurable subset A K: P (f k (U ) A) P (f(u [a,b) ) A) < 1/k Finally, by Theorem. we know, that the distribution of Y = f(u [a,b) ) is the Young measure assoiated with the funtion f. Thus we an formulate the following result onerning lassial Young measures. Proposition 3.1: Classial Young measure generated by ROSU(f) is the homogeneous Young measure. This measure is idential with the distribution of the random variable Y = f(u [a,b) ). IV. THEORETICAL APPLICATIONS On the basis of the above general haraterization of the lassial Young measure generated by ROSU(f) we an obtain a number of rules whih allow one to find an explit form of the lassial Young measure in various speifi ases. For example, let us onsider the following situation. Let [a, b) be the interval-domain of the funtion f. Let us onsider an open partition of [a, b) into a number of open intervals I 1, I,..., I n suh that the intervals are pairwise disjoint and n I i = [a, b], where A denotes the losure of i the set A. Let funtion f be ontinuously differentiable on eah interval of the partition and let f (x) for all x n I l l=1 Using the well-known probabilisti result onerning the distributions of suh funtions of random variables we an obtain the following orollary of the Proposition 3.1. Corollary.1: The lassial Young measure generated by any ROSU(f) is a homogeneous one and its density g with respet to the Lebesgue measure on K is of the following form g(y) = 1 b a n h l(y) 1 Dl (y) () l=1 where h l is the inverse of f on the interval I l, while D l = f(i l ) is the domain of h l. The symbol 1 stands for the haratertisti funtion of the set indiated in its subsript. V. APPLICATIONS TO COMPUTATION OF YOUNG FUNCTIONAL S VALUES To show in what way the above theoretial results work in pratie, let us onsider a ase where both sides of the Eq. 1 an be omputed preisely. Example 1 - ontinuation As previously let us onsider the funtion (3) that is defined on the interval [, 13/) and the ROSU(f) that is defined on the interval [, 5). Now our aim is to ompute (Advane online publiation: 7 November 18)

4 IAENG International Journal of Applied Mathematis, 8:, IJAM_8 3 (a) (b) () Fig. 1: A funtion f given by Eq. (3) and funtions f 1, f 3 and f 35 belonging to ROSU(f) with the domain = [, 5). Plots of these funtions are labeled as a, b,, and d, respetively (d) the integrals C(f k, H) that appear on the left-hand side of the Eq. 1 for the exemplary Carathéodory funtion H(x, y) = x y. Fig. provides us with the graphial illustration of the C(f 3 ; H) and C(f 5 ; H) in this ase. The values of these integrals are the areas of the surfaes presented in plot (a) and (b), respetively. In order to ompute the limit value in Eq. 1 let us note, that in the onsidered ase where A k C(f k, H) B k (5) A k = 197(169[k/13] 585[k/13] + 98[k/13] 3 )/183k 3 B k = 197( [k/13] + 355[k/13] + 98[k/13] 3 )/183k 3 (6) with [x] denoting the greatest integer less than or equal to x. Beause the limit of both A k, B k as k tends to infinity is the same and equales L1 = 615/1, it is also the limit of C(f k, H) that we have been looking for. Now let us find the limit with the help of the Eq. 1. i.e. by omputation of the value of the Young funtional YF(H). For this purpose we need to know the lassial Young measure generated by the ROSU(f). With the help of Corollary.1 one an easily reeive the following formula for its density funtion: g(y) = /(13 y)1 [,5/16) (y) +/(13 y)1 [5/16,) (y) Thus the value of the Young funtional in the onsidered ase is the following (reall that the Young measure ν x is homogeneous in this ase, so we an omit subsript x in the denotations) : [,5) [,) YF(H) = H(x, y)dν(y)dx = (7) H(x, y)g(y)dydx = 5/16 K 5/16 x y 13 dydx + (8) y x y dydx = = L1 (9) y 1 So, as it should be expeted, the whole information about the rapid osillations in this ase is ontained in the lassial Young measure and - due to the Proposition an be revealed with the help of the Corollary.1. Beause we are going to make use of this example as a kind of benhmark in our next onsiderations, let us also ompute both the limit and the Young funtional values for another Carathéodory funtion, namely let H(x, y) = x os(y). In this ase little alulus shows that C(fk; H) satisfy (5) with A k = [k/13] 16k (3 (F ( /) + F (5/( )))+ 13 [k/13](f ( /) + F C(5/( ))+ 8(sin(5/16) sin())) B k = 1+[k/13] 16k (13 [k/13](f ( /)+ F (5/( )) + 16 (F ( /)+ F (5/( )) + 8(sin(5/16) sin())) (1) where F denotes the speial funtion known as the Fresnel (Advane online publiation: 7 November 18)

5 IAENG International Journal of Applied Mathematis, 8:, IJAM_8 3 (a) Fig. : Graph of two exemplary surfaes - plot (a) and plot (b) - the area of whih are, respetively, the values of the integrals C(f 3 ; H) and C(f 5 ; H). These integrals are elements of the sequene on the left-hand side of the Eq. 1 (b) funtion that is given by the following integral: F (z) = z os(t /)dt Again, the limits of both A k, B k are the same, so they are the same as the limit of C(f k, H) in this ase. It equals L = (5 /13)(F ( /) + F (5/( )) (11) The same value an be obtain with the help of our result. Indeed, it is easy to hek, that the integral (7) in the onsidered ase takes on the form: 5/16 x os(y) 13 y dydx + 5/16 x os(y) 13 y dydx and it is equal to L. Our example is relatively simple and smooth - we were able to ompute the limits expliitly. However even here one an notie the benefits resulting from our Proposition. It is quite lear that even in this ase where the osillations have suh smooth nature, the omputation of the limit of C(f k, H) for more omplex Carathéodory funtions ould be muh more diffiult task than the alulation of the value of the Young funtional YF(H). The possibility of derivation of expliit formulae for the density funtions of lassial Young measures is not the only benefit resulting from our Proposition 3.1. In many interesting ases expliit formulae for these densities annot be found. For instane, if one wants to make use of Corollary.1 they have to obtain the inverses of f on partiular subintervals of its domain, but it is not always possible. However in all suh ases thanks to Proposition 3.1 one may use diretly Monte Carlo simulations in order to evaluate values of the Young funtionals. The idea of the Monte Carlo simulation is well-known. Its main onept is to draw a sample Z 1, Z,..., Z m of realizations of independent random variables with the same distribution as the random phenomenon under study. Based on this sample, important information onerning stohasti harateristis of the examined distribution an be derived with the help of statistial-inferene tools. Monte Carlo simulations are also frequently used to evaluate values of integrals in various situations. Beause now we onsider ases where the lassial Young measure is not given expliitly, we adopt this approah to evaluate Young funtional d ( b YF(H) = H(x, y)dν(y) ) dx. When the hange of a integration order is possible, then this integral is equal to b ( d YF(H) = H(x, y)dx) ) dν(y). a By Proposition 1, the Young measure ν is the probability distribution of the random variable Y=f(U(a,b)). So, we may look at the Young-funtional-value as the expeted value d of the random variable H(x, Y )dx and it is well-known that it an be best-estimated by its empirial mean (i.e. the arithmeti mean of its randomly generated values). Thus we propose to make use of the following proedure YFE(H, f, a, b,, d, N) to evaluate values of the Young funtional: Step : Set k = 1 Step 1: Set t=random((a,b)) Step : Set y=f(t) Step 3: Set z[k]=int(h(x,y),x,,d) Step : N times repeat Steps 1 to 3 Step 5: Set sample=(z[1],...,z[n]) Step 6: Set YF=Mean(sample) Step 7: Return YF The above proedure is alled with the following arguments: the formula that defines the Caratheodory funtion H, the formula for the funtion f that defines the ROSU and its domain, i.e. the interval [a,b), the domain [,d) of the ROSU itself, and the size N of the sample that will be used to evaluate the Young funtional value. The subroutine Random(I) returns a pseudorandom number generated aording to the uniform probability distribution defined on an interval I. The subroutine INT(H(x,y), x,,d) returns the value (Advane online publiation: 7 November 18)

6 IAENG International Journal of Applied Mathematis, 8:, IJAM_8 3 TABLE I: Results Of Monte Carlo Evaluations of the Young Funtional Values for Problems Considered in Example 1 Caratheodory Sample Estimated Relative integrand size value error H(x, y) = x y H(x, y) = x os(y) of the integral d H(x, y)dx. Obviously the integration an be done numerially, beause when the subroutine is alled, the value of y is already fixed, as it is set up at the Step. The usefulness of the Monte Carlo simulations in omputation of the values of the Young funtionals is supported by the data presented in Table I. As one an see, the auray of the Monte Carlo evaluations are realy very good (let us reall that the true values are L1 and L, given by (9) and (11), respetively). Other examples of the advantages resulting from the Monte Carlo approah to omputation of Young funtionals values an be found in [8]. The pseudoode of the proedure YFE presented in this setion an be implemented in various ways. In our researh we use the following version oded in Wolfram Mathematia 1.5 programming environment: Mean[Table[ NIntegrate[H[x,f[RandomReal[{a,b}]]], {x,,d}],nn]] where NN is the sample size, while the remaining symbols have an analogous meaning as in the desription of the YFE. Now, let us onsider the following funtion G: G(y) = d H(x, y)dx (1) If the Caratheodory integrands are suh that the funtion G has got known open formula valid for all y (a, b), then a faster version of YFE an be obtain by hanging the original Step 3 in the following way: Step 3: Set z[k]=g(y) In our exemplary problems, it an be atually done, so we made use of the following Wolfram Mathematia ode: Mean[Table[G[f[RandomReal[{a,b}]]]],NN]] where, for the problems disussed in Example 1 G(y) = 15y/3 if H(x, y) = x y and G(y) = 5 os(y)/ if H(x, y) = x os(y). Reall that in both ases the limits in integral (1) are =, d = 5 We see that pratial implementation of presented Monte Carlo evaluation proedure is very simple and taking into aount the auray of the values we have obtained, it is also very ompetitive alternative tool for engineers. As a onsequene, various different rules for omputing expliit formulae for the density funtions of lassial Young measures generated by ROSU(f) an also be obtained on the basis of the result stated in Proposition 3.1. We are also sure that the same approah enables development of analogous results related to rapidly osillating sequenes with uniform representation whih are defined on open and bounded subsets of R d. In our opinion it is very promising diretion of future researh. REFERENCES [1] J.B. Baani, and G. Peihl,,,The Seond-Order Eulerian Derivative of a Shape Funtional of a Free Boundary Problem, IAENG International Journal of Applied Mathematis, vol. 6, no., pp.5 36, 16. [] J.M. Ball,,,Mathematial Models of Martensiti Mirostruture, Materials Siene and Engineering A 378. pp ,. [3] J.M. Ball, and R.D. James, Fine Phase Mixtures as Minimizers of Energy, Arhive for Rational Mehanis and Analysis, 1, pp.13 5, [] P.Billingsley, Convergene of Probability Measures, John Willey & Sons, [5] M. Chipot, and C. Collins, and D. Kinderlehrer,,,Numerial Analysis of Osillations in Multiple Well Problems, Numerishe Mathematik 7, pp 59 8, [6] A.Z. Grzybowski, and P. Puhała, On general haraterization of Young measures assoiated with Borel funtions, preprint, arxiv: 161.6v, submitted. [7] A.Z. Grzybowski, and P. Puhała, On Classial Young Measures Generated by Certain Rapidly Osillating Sequenes, Leture Notes in Engineering and Computer Siene: Proeedings of The World Congress on Engineering and Computer Siene 17, 5-7 Otober, 17, San Franiso, USA, pp [8] A.Z. Grzybowski, and P. Puhała, Monte Carlo Simulation in the Evaluation of the Young Funtional Values, Proeedings of IEEE 1th International Sientifi Conferene on Informatis, Poprad, (eds. Novitzka, S. Koreko, A. Szakal), New York 17, pp1-6 [9] W. Han, and B.D. Reddy,,,Computational Plastiity: the Variational Basis and Numerial Analysis, Computational Mehanis Advanes, pp. 83, [1] G. Hou, and G. Wang, and Z. Pan, and B. Huang, and H. Yang, and T. Yu, Image Enhanement and Restoration: State of the Art of Variational Retinex Models, IAENG International Journal of Computer Siene, vol., no., pp5 55, 17. [11] S. Müller, Variational Models for Mirostruture and Phase Transitions, in Calulus of variations and geometri evolution problems, Hildebrandt, S., Struwe, M. (editors), Leture Notes in Mathematis Volume 1713, Springer Verlag, Berlin Heidelberg, Germany [1] P. Puhała, An elementary method of alulating Young measures in some speial ases, Optimization, vol. 63 no.9, pp , 1. [13] P. Pedregal, Variational Methods in Nonlinear Elastiity, SIAM, Philadelphia. [1] T. Roubíček, Relaxation in Optimization Theory and Variational Calulus, Walter de Gruyter, Berlin, New York, [15] L.C. Young, Generalized urves and the existene of an attained absolute minimum in the alulus of variations, Comptes Rendus de la Soiété des Sienes et des Lettres de Varsovie, lasse III vol. 3, pp. 1 3, [16] L.C. Young, Generalized surfaes in the alulus of variations, Ann. Math. vol. 3, part I: pp. 8 13, part II: pp. 53 5, 19. VI. FINAL REMARK The probability theory provides us with a number of different versions of the theorems onerning the probability distributions of funtions of random variables and/or vetors. (Advane online publiation: 7 November 18)