SOBOLEV SPACE OF FUNCTIONS VALUED IN A MONOTONE BANACH FAMILY NIKITA EVSEEV AND ALEXANDER MENOVSCHIKOV Abstract. We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian spaces. For a monotone family, we show the existence of weak derivative, obtain an isomorphism to the standard Sobolev space, and provide some scalar characteristics. 1. Introduction Various applied problems in biology, materials science, mechanics, etc, involve PDEs with solution spaces with internal structure that changes over time. As examples, we review some of the recent researches: [1] by M. L. Bernardi, G. A. Pozzi, and G. Savar´e,[2] by F. Paronetto (equations on non-cylindrical domains), [3] by M. Vierling, [4] by A. Alphonse, C. M. Elliott, and B. Stinner (equations on evolving hypersurfaces), [5] by S. Meier and M. B¨ohm,[6] by J. Escher and D. Treutler (modeling of processes in a porous medium). Common to all of the above examples is that solution spaces could be represented as sets of functions valued in a family of Banach spaces. However, different problems impose different requirements on the relations between spaces within families, for example, the existence of isomorphisms, embeddings, bounded operators and so on. In this article, we consider Sobolev spaces associated with the above problems from the point of view of metric analysis. Although the family of Banach spaces cannot always be represented as a metric space, the metric definition of Sobolev classes remains meaningful. Such a point of view on the studied spaces will make it possible to apply more universal and well-developed methods. In the 90s, several authors (L. Ambrosio [7], N.J. Korevaar and R.M. Schoen [8], Yu.G. Reshetnyak [9] and A. Ranjbar-Motlagh [10]) introduced and studied Sobolev spaces consisting of functions taking values in metric spaces. The case of functions defined on a non-Euclidean space is described by P. Haj lasz [11] and J. Heinonen, P. Koskela, arXiv:2003.10657v1 [math.FA] 24 Mar 2020 N. Shanmugalingam, J.T. Tyson [12]. In [12] it was shown that all previously developed approaches are equivalent. For a detailed treatment and for references 2010 Mathematics Subject Classification. 46E40, 46E35. Key words and phrases. Sobolev spaces of vector-valued functions, Lp-direct integral, Bochner integral. The work was supported by Russian Foundation for Basi Resear h (proje t no. 18-31-00089). 1 2 to the literature on the subject one may refer to books [13] by J. Heinonen and [14] by P. Haj laszand P. Koskela. For our purposes, we adapt the following definition of Sobolev space (Newtonian spaces, for real-valued case see [15], and [12] for Banach-valued case). Function f :(M; ρ) ! (N; d) from the space Lp(M; N) belongs to W 1;p(M; N), if there exists scalar function g 2 Lp(M) such that Z df(γ(a)); f(γ(b)) ≤ sup g dσ: (1.1) γ γ On the one hand, we have all the necessary objects to adapt this definition. On the other hand, in metric case, it allows us to introduce the concept of an upper gradient, establish the Poincar´einequality, show that a superposition with a Lipschitz function also is a Sobolev function. The evolution structure of a specific problem could be described with the help of the following objects. Let fXtgt2(0;T ) be a family of Banach spaces, (0;T ) ⊂ R, and suppose that there is a set of operators P (s; t): Xs ! Xt for t ≥ s. We consider functions with the property that f(t) 2 Xt. Then inequality (1.1) turns to Z t kf(t) − P (s; t)f(s)kt ≤ g(τ) dτ; s 1;p and defines the space W ((0;T ); fXtg). The first natural question arises from this definition is: what is the meaning of the function g(t)? In the case of a monotone family of reflexive spaces, the answer to this question is given by theorem 4.5. Namely, under such assumptions, we can explicitly construct the weak derivative and show that its norm coincides with the smallest upper gradient of the original function. In section 5 we establish the connection of the introduced space to the stan- dard case. More precisely, suppose that there is a set of local isomorphisms Φt : Xt ! Y . We are interested if there exists a global isomorphism between 1;p 1;p Sobolev spaces W ((0;T ); fXtg) and W ((0;T );Y ). Due to theorem 5.4, the necessary and sufficient conditions for the existence of such an isomorphism are the close interconnection of Φt and the transition operators P (s; t). In section 6, as an example, we give a comparison of our approach with that of Yu.G. Reshetnyak (recall that for metric spaces both came up with the same re- sults). It turns out that Reshetnyak's approach does not allow taking into account the internal structure of the spaces under consideration. 1;p 2. Sobolev space W ((0;T ); fXtg) In this section, we give the definition of the main object - Sobolev functions valued in the family of Banach spaces. We also provide examples of families on which our methods can be applied. 3 2.1. Monotone family fXtg as a vector space. Let V be a vector space, (0;T ) be an interval (not necessarily bounded) equipped with the Lebesgue measure, and fjj · jjtgt2(0;T ) be a family of semi-norms on V . We will assume that for each v 2 V function N(t; v) = kvkt is non-increasing: N(t1; v) ≥ N(t2; v); if t1 ≤ t2: (2.1) Define Banach space Xt to be a completion V= ker(k·kt) with respect to k·kt. Then fXtg is said to be a monotone family of Banach spaces (or a monotone Banach family). For t1 ≤ t2 define operators P (t1; t2): Xt1 ! Xt2 in the following way. If x 2 Xt1 then there is a sequence fvkg ⊂ V such that it is a Cauchy sequence with respect to k · kt1 and x is its limit in Xt1 . Due to (2.1) fvkg is a Cauchy sequence with respect to k · kt2 , thus there is its limitx ~ in Xt2 . Put P (t1; t2)x =x ~. Now we can construct an addition. If xi 2 Xti then ( P (t1; t2)x1 + x2; if t1 ≤ t2; x1 + x2 := (2.2) x1 + P (t2; t1)x2; if t1 > t2: One can show that this operation satisfy the associative and commutative proper- S ties. So the pare ( t Xt; +) is a vector space over R. 2.2. Lp-direct integral of Banach spaces. We deal with the Lp-spaces of map- S pings f : (0;T ) ! t Xt with the property that f(t) 2 Xt for each t 2 (0;T ) (in other words, f is a section of fXtg ). To make this treatment rigorous, we apply the concept of direct integral of Banach spaces. A brief account of the theory of direct integral is given below (for detailed presentation see [16] and [17]). Note that monotonicity condition (2.1) implies that fXtg is a measurable family of Banach spaces over ((0;T ); dt; V ) in the sense of [16, Section 6.1]. Definition 2.1. A simple section is a section f for which there exist n 2 N, Pn v1; : : : ; vn 2 V , and measurable sets A1;:::;An ⊂ T such that f(t) = k=1 χAk ·vk for all t 2 (T; 0). Definition 2.2. A section f of fXtgt2(0;T ) is said to be measurable if there exists a sequence of simple sections ffkgk2N such that for almost all t 2 (0;T ), fk(t) ! f(t) in Xt as k ! 1. The space of all equivalence classes of such measurable sections is a direct integral R ⊕ (0;T ) Xt dt of a monotone family of Banach spaces fXtgt2(0;T ) We will denote this 0 space as L ((0;T ); fXtg). Note that for a measurable section f the function t 7! kf(t)kt is measurable in p R ⊕ the usual sense. For every p 2 [1; 1] the space L ((0;T ); fXtg) = (0;T ) Xt dt Lp (Lp-direct integral) is defined as a space of all measurable sections f such that the 4 p function t 7! kf(t)kt belongs to L ((0;T )). In this case 8 1 p > R T p < 0 kf(t)kt dt ; if p < 1; p kfkL ((0;T );fXtg) := >ess sup jf(t)j; if p = 1 : (0;T ) p determines the norm on L ((0;T ); fXtg). p Proposition 2.3 ( [17, Proposition 3.2]). The space L ((0;T ); fXtg) is a Banach space for all 1 ≤ p < 1. 0 Define sectional weak convergence fn(t) * f(t) for a.e. t 2 (0;T ) as hb (t); fn(t)it ! 0 0 0 hb (t); f(t)it a.e. for all b (t) 2 Xt. Then applying a standard technique one can prove the following proposition. p p Proposition 2.4. Let fn 2 L ((0;T ); fXtg), kfnkL ((0;T );fXtg) ≤ C < 1 and p p fn(t) * f(t) for a.e. t 2 (0;T ). Then f 2 L ((0;T ); fXtg) and kfkL ((0;T );fXtg) ≤ C. 1;p 2.3. Sobolev space W ((0;T ); fXtg). As it was pointed out in the introduction, we adapt (1.1) and obtain the definition of Newtonian space for functions valued in a monotone family.