On the distributional divergence of vector fields vanishing at infinity

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1 Proceedings of the Royal Society of Edinburgh, 141A, 65 76, 2011 On the distributional divergence of vector fields vanishing at infinity Thierry De Pauw Institut de Recherches en Mathématiques et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium Monica Torres Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN , USA (MS received 31 August 2009; accepted 1 April 2010) The equation div v = F has a solution v in the space of continuous vector fields vanishing at infinity if and only if F acts linearly on BV m/(m 1) ( ) (the space of functions in L m/(m 1) ( ) whose distributional gradient is a vector-valued measure) and satisfies the following continuity condition: F (u ) converges to zero for each sequence {u } such that the measure norms of u are uniformly bounded and u 0 weakly in L m/(m 1) ( ). 1. Introduction The equation u = f L m ( ) need not have a solution u C 1 ( ). In this paper we prove that, to each f L m ( ), there corresponds a continuous vector field, vanishing at infinity, v C 0 ( ; ) such that div v = f weakly. In fact, we characterize those distributions F on such that the equation div v = F admits a weak solution v C 0 ( ; ). Related results have been obtained in [1 4, 6]. Our first proof, contained in 3 6, follows the same pattern as [2]. A second proof, presented in 7, is based on the more abstract methods developed in [3]. In this paper m 2 and 1 := m/(m 1). Let BV 1 ( ) denote the subspace of L 1 ( ) consisting of those functions u whose distributional gradient u is a vector-valued measure (of finite total mass). We define a charge vanishing at infinity to be a linear functional F :BV 1 ( ) R such that F (u ) 0 whenever u 0 weakly in L 1 ( ) and sup u M <. (1.1) We denote by CH 0 ( ) the space of charges vanishing at infinity and we note (see proposition 3.2) that it is a closed subspace of the dual of BV 1 ( ) (where the latter is equipped with its norm u M ). Examples of charges vanishing at infinity include the functions f L m ( ) (see proposition 3.4) and the distributional 65 c 2011 The Royal Society of Edinburgh

2 66 Th. De Pauw and M. Torres divergence div v of v C 0 ( ; ) (see proposition 3.5). Our main result thus consists in proving that the operator C 0 ( ; ) CH 0 ( ): v div v (1.2) is onto. This is done by applying the closed range theorem. For this purpose we identify CH 0 ( ) with BV 1 ( ) via the evaluation map (see proposition 5.1). This in turn relies on the fact that L m ( ) is dense in CH 0 ( ) (see corollary 4.3, which is obtained by smoothing). Therefore, the adoint of (1.2) is BV 1 ( ) M( ; ): u u. The observation that this operator has a closed range follows from compactness in BV 1 ( ) (see proposition 2.6). Charges vanishing at infinity happen to be the linear functionals on BV 1 ( ) which are continuous with respect to a certain locally convex linear (sequential, nonmetrizable, non-barrelled) topology T C on BV 1 ( ). In other words, there exists a locally convex topology T C on BV 1 ( ) such that a sequence u 0 in the sense of T C if and only if the sequence {u } verifies the conditions of (1.1). Topologies of this type have been studied in [3, 3]. Referring to the general theory yields a quicker, though very much abstract proof in 7. In order to appreciate this alternative route, the reader is expected to be familiar with the methods of [3, 3]. From this perspective the key identification CH 0 ( ) = BV 1 ( ) is simply saying that BV 1 ( )[T C ] is semireflexive; a property which follows from the compactness proposition Preliminaries A continuous vector field v : is said to vanish at infinity if, for every ε>0, there exists a compact set K such that v(x) ε whenever x \ K. These form a linear space denoted by C 0 ( ; ), which is complete under the norm v := sup{ v(x) : x }. The linear subspace C c ( ; ) (respectively, D( ; )) consisting of those vector fields having compact support (respectively, smooth vector fields having compact support) is dense in C 0 ( ; ). Thus, each element of the dual, T C 0 ( ; ), is uniquely associated with some vectorvalued measure µ M( ; ) as follows: T (v) = v, dµ, according to the Riesz Markov representation theorem. Furthermore, { } µ M = sup v, dµ : v D( ; ) and v 1. A vector-valued distribution T D( ; ) with the property that sup{t (v): v D( ; ) and v 1} < extends uniquely to an element of C 0 ( ; ) and is therefore associated with a vector-valued measure as above.

3 Distributional divergence 67 We recall some properties of convolution. Let 1 p<, u L p ( ) and ϕ D( ). For each x, we define (u ϕ)(x) = u(y)ϕ(x y)dy. It follows from Young s inequality that u ϕ L p ( ) and u ϕ L p u L p ϕ L 1. (2.1) Furthermore, u ϕ C ( ) and (u ϕ) =u ϕ. In the case when ϕ is even and f L q ( ) with p 1 + q 1 =1,wehave f(u ϕ) = u(f ϕ). We fix an approximate identity on, {ϕ k } [5, (6.31)], and we infer that lim k u u ϕ k L p =0. (2.2) Henceforth we assume that m 2. We let the Sobolev conugate exponent of 1 be 1 := m m 1. Note that L 1 ( ) is isometrically isomorphic to L m ( ). We will recall the Gagliardo Nirenberg Sobolev inequality whenever ϕ D( ). ϕ L 1 κ m ϕ L 1 Definition 2.1. We let BV 1 ( ) denote the linear subspace of L 1 ( ) consisting of those functions u whose distributional gradient u is a vector-valued measure, i.e. { } u M = sup u div v : v D( ; ) and v 1 <. Readily u := u L 1 + u M defines a norm on BV 1 ( ), which makes it a Banach space. In view of proposition 2.5, we will use the equivalent norm u BV1 := u M. Definition 2.2. Given a sequence {u } in BV 1 ( ), we write u 0 whenever (i) sup u M <, (ii) u 0 weakly in L 1 ( ). Proposition 2.3. Let {u } be a sequence in BV 1 ( ), u L 1 ( ), and assume that u uweakly in L 1 ( ). It follows that u M lim inf u M. (2.3)

4 68 Th. De Pauw and M. Torres Proof. Let v D( ; ) with v 1. Since div v L m ( ) and u weakly in L 1 ( ) we have, from definition 2.1, u div v = lim u div v lim inf u M and, taking the supremum over all such v, we conclude that u u M lim inf u M. The following density result is basic. Proposition 2.4. Let u BV 1 ( ). The following hold: (i) for every ϕ D( ), u ϕ BV 1 ( ) and (u ϕ) L 1 u M ϕ L 1; (ii) if {ϕ k } is an approximate identity, then u u ϕ k 0 and lim (u ϕ k ) L 1 = u M ; k (iii) there exists a sequence {u } in D( ) such that u u 0 as well as lim u L 1 = u M. Proof. We note that (2.1) yields u ϕ L 1.Wehave (u ϕ) (x)dx = ϕ u (x)dx R m = ϕ(x y)d u(y) dx R m ϕ(x y) d u (y)dx R ( m ) = ϕ(x y) dx d u (y) = u M ϕ L 1, (2.4) which shows proposition 2.4(i). Let {ϕ k } be an approximate identity. From proposition 2.4(i), we obtain (u ϕ k ) M = (u ϕ k ) (x)dx u M ϕ k L 1 = u M. (2.5) Since u ϕ k u in L 1 ( ), then, in particular, u ϕ k uweakly in L 1 ( ); i.e. f[(u ϕ k ) u] 0 for every f L m ( ). (2.6)

5 Distributional divergence 69 From (2.5) and (2.6) we obtain that u u ϕ k 0. Moreover, from (2.5) and the lower semicontinuity (2.3) we conclude that lim k (u ϕ k ) L 1 = u M, which shows that proposition 2.4(ii) holds. In order to establish (iii), we choose a sequence {ψ i } in D( ) such that 1 B(0,i) ψ i 1 B(0,2i) and sup ψ i L m <. (2.7) i As usual, let {ϕ k } be an approximate identity. Referring to proposition 2.4(ii) we inductively define a strictly increasing sequence of integers {k } such that (u ϕ k ) u M + 1 R. m For each and i, we observe that [(u ϕ k )ψ i ] ψ i (u ϕ k ) + (u ϕ k ) ψ i. For fixed we infer from (2.7) and the relation u ϕ k 1 L 1 ( ) that lim sup (u ϕ k ) ψ i = lim sup (u ϕ k ) ψ i i i B(0,i) c ( ) 1/1 lim sup u ϕ k 1 ψ i L m i B(0,i) c =0. According to the three preceding inequalities we can define inductively a strictly increasing sequence of integers {i } such that [(u ϕ k )ψ i ] (u ϕ k ) + 1 R u M + 2. m We set u := (u ϕ k )ψ i. In view of proposition 2.3, it only remains to show that u uweakly in L 1 ( ). Given f L m ( ), we note that f(u (u ϕ k )ψ i ) f u (u ϕ k ) + f u ϕ k 1 ψ i ( ) 1/m f L m u (u ϕ k ) L 1 + f m u ϕ L1 k L 1. B(0,i ) c The latter tends to zero as and the proof is complete. Proposition 2.5 (Gagliardo Nirenberg Sobolev inequality). Let u BV 1 ( ). We have u L 1 κ m u M. Proof. Since the norm L 1 in L 1 ( ) is lower semicontinuous with respect to weak convergence, the result is a consequence of proposition 2.4(iii) and the Gagliardo Nirenberg Sobolev inequality for functions in D( ).

6 70 Th. De Pauw and M. Torres Proposition 2.6 (compactness). Let {u } be a bounded sequence in BV 1 ( ), i.e. sup u M <. Then there exist a subsequence {u k } of {u } and u BV 1 ( ) such that u k u 0. Proof. Since {u } is bounded in BV 1 ( ), it is also bounded in L 1 ( ) according to proposition 2.5. The conclusion thus immediately follows from the fact that L 1 ( ) is a reflexive Banach space whose dual is separable, together with proposition Charges vanishing at infinity Definition 3.1. A charge vanishing at infinity is a linear functional F :BV 1 ( ) R such that u,f 0whenever u 0. The collection of these is denoted by CH 0 ( ). We readily see that CH 0 ( ) is a linear space. With F CH 0 ( ) we associate F CH0 := sup{ u, F : u BV 1 ( ) and u M 1}. We check that F CH0 < for each F CH 0 ( ) according to proposition 2.6; hence CH0 is a norm on CH 0 ( ). Note that CH 0 ( ) BV 1 ( ) and F CH0 = F (BV1 ) whenever F CH 0( ). Proposition 3.2. CH 0 ( )[ CH0 ] is a Banach space. Proof. Let {F k } be a Cauchy sequence in CH 0 ( ). It follows that {F k } converges in BV 1 ( ) to some F BV 1 ( ) and it remains only to check that F is a charge vanishing at infinity. Let {u } be a sequence in BV 1 ( ) such that u 0 and put Γ := sup u M. Given ε>0, choose an integer k such that F F k BV 1 ε. Observe that, for each, u,f u,f k + u,f F k u,f k + F F k BV Γ 1 u,f k + εγ. Thus, lim sup u,f εγ, and since ε is arbitrary the conclusion follows. The following is a ustification for the terminology vanishing at infinity. Proposition 3.3. Let F CH 0 ( ) and ε>0. Then there exists a compact set K such that u, F ε u M whenever u BV 1 ( ) and K supp u =. Proof. Let F CH 0 ( ). Assume, if possible, that there exist ε>0and a sequence {u } in BV 1 ( ) such that u M =1,B(0,) supp u =, and u,f ε for every. We claim that u 0. In order to show this, it suffices to establish that u 0 weakly in L 1 ( ). Let f L m ( ). Given η>0, there exists a compact set K such that f m η m. \K

7 Distributional divergence 71 If is sufficiently large for K B(0,), then ( ) 1/m fu = fu f m u L 1 ηκ m. \K \K Thus, lim sup fu ηκ m and, since η is arbitrary, we infer that fu 0. This establishes our claim and in turn implies that lim u,f = 0, which is a contradiction. We now turn to giving the two main examples of charges vanishing at infinity. Given f L m ( ) (and recalling that BV 1 ( ) L 1 ( )), we define Λ(f): BV 1 ( ) R: u uf. Proposition 3.4. Given f L m ( ), we have Λ(f) CH 0 ( ) and Thus, is a bounded linear operator. Λ(f) CH0 κ m f L m. Λ: L m ( ) CH 0 ( ) Proof. Let {u } be a sequence in BV 1 ( ) such that u 0. Then u 0 weakly in L 1 ( ), whence u,λ(f) 0, thereby showing that Λ(f) CH 0 ( ). Given u BV 1 ( ), we note that so that Λ(f) CH0 κ m f L m. u, Λ(f) u L 1 f L m κ m u M f L m Given v C 0 ( ; ) and u BV 1 ( ), we note that v is summable with respect to the measure u. Thus, we may define Φ(v): BV 1 ( ) R: u v, d( u). Proposition 3.5. Given v C 0 ( ; ), we have Φ(v) CH 0 ( ) and Thus, is a bounded linear operator. Φ(v) CH0 v. Φ: C 0 ( ; ) CH 0 ( )

8 72 Th. De Pauw and M. Torres Proof. Let v C 0 ( ; ) and let {u } be a sequence in BV 1 ( ) such that u 0. Given ε>0, we choose w D( ; ) such that w v ε. Set Γ = sup u M. We note that u,φ(v) v w, d( u ) + u div w R R εγ + u div w R. m m m Since supp div w is compact, we infer that div w L m ( ). Hence, lim u div w =0. Thus, lim sup u,φ(v) εγ and, from the arbitrariness of ε, we conclude that Φ(v) CH 0 ( ). Finally, if u BV 1 ( ), then u, Φ(v) = v, d( u) v u M, and thus Φ(v) CH0 v. 4. Approximation Let F CH 0 ( ) and ϕ D( ). Our goal is to define a new charge vanishing at infinity, the convolution of F and ϕ, denoted by F ϕ, to show that it belongs to the range of Λ (see proposition 3.4), and that it approximates F in the norm CH0. We start by observing that if u BV 1 ( ), then u ϕ BV 1 ( ) (see proposition 2.4(i)). Therefore, F ϕ: BV 1 ( ) R: u u ϕ, F is a well-defined linear functional. We now show that F ϕ is indeed a charge vanishing at infinity, in fact, of the special type Λ(f) for some f L m ( ). We denote by R(Λ) the range of the operator Λ. Proposition 4.1. Let F CH 0 ( ) and ϕ D( ). It follows that F ϕ CH 0 ( ) R(Λ). Proof. The restriction of F to D( ) is a distribution, still denoted by F. Thus, the convolution F ϕ is associated with a smooth function f C ( ) as follows: ψ, F ϕ = ψf (4.1) for every ψ D( ) (see, for example, [5, (6.30b)]). We claim that f L m ( ). Let {ψ } be a sequence in D( ) such that ψ L 1 0. Note that sup (ψ ϕ) M = sup (ψ ϕ) L 1 sup ψ L 1 ϕ L q <, where q = m/(m + 1), according to Young s inequality. For any g L m ( ), we have g(ψ ϕ) = ψ (g ϕ) 0,

9 Distributional divergence 73 since g ϕ L m ( ) and ψ 0 weakly in L 1. Therefore, ψ ϕ 0 and, in turn, ψ ϕ, F = ψ,f ϕ 0. This shows that F ϕ is L 1 -continuous in D( ). Since D( ) is dense in L 1 ( ), we infer that F ϕ can be uniquely extended to a continuous linear functional on L 1 ( ). Thus, the Riesz representation theorem yields f L m ( ) and, therefore, proposition 3.4 gives Λ(f) CH 0 ( ). It only remains to show that Λ(f) = F ϕ, which is equivalent to showing that (4.1) actually holds for every ψ BV 1 ( ). To see this, we use proposition 2.4(iii) to obtain a sequence {ψ } D( ) such that ψ ψ. Note that equation (4.1) holds for each ψ, and the result follows by noting that ψ f ψf and ψ,f ϕ = ψ ϕ, F ψ ϕ, F = ψ, F ϕ, since ψ ψweakly in L 1 ( ) and ψ ϕ ψ ϕ. It remains to show that F ϕ is a good approximation of F in CH 0 ( ) provided that ϕ is a good approximation of the identity. Proposition 4.2. Let F CH 0 ( ) and let {ϕ k } be an approximate identity such that each ϕ k is even. It follows that lim k F F ϕ k CH0( ) =0. Proof. In order to simplify the notation we put F k = F ϕ k. Since F CH 0 ( ), the following holds. For every ε>0, there are f 1,...,f J L m ( ) and positive real numbers η 1,...,η J such that u, F ε whenever u BV 1 ( ), u M 2 and uf η for every =1,...,J. We associate an integer k with each =1,...,J such that f f ϕ k L m η κ m whenever k k. Now, given u BV 1 ( ) with u M 1, and given k max{k 1,...,k J }, we infer that (u u ϕ k ) M 2 and, for each =1,...,J, (u u ϕ k )f = uf (u ϕ k )f R m R m = uf u(f ϕ k ) R m = u(f f ϕ k ) Therefore, u L 1 f f ϕ k L m η. u, F F k = u u ϕ k,f ε.

10 74 Th. De Pauw and M. Torres Taking the supremum over all such u, we obtain F F k ε whenever k max{k 1,...,k J }, and the proof is complete. Corollary 4.3. R(Λ) is dense in CH 0 ( ). 5. Duality Proposition 5.1. The evaluation map ev : BV 1 ( ) CH 0 ( ) is a biection. Proof. Since F, ev(u) = u, F, we readily infer that ev is inective. We now turn to proving that ev is surective. Let α CH 0 ( ). It follows from proposition 3.4 that α Λ (L m ( )). Thus, there exists u L 1 ( ) such that Λ(f),α = uf for every f L m ( ). (5.1) Given v D( ; ), we note that the charges Φ(v) (see proposition 3.5) and Λ(div v) (see proposition 3.4) coincide, according to proposition 2.4(iii), because they trivially coincide on D( ). Thus, u div v = Λ(div v),α = Φ(v),α α CH 0 Φ(v) CH0 α CH 0 v according to proposition 3.5. This proves that u BV 1 ( ). It then follows from (5.1) that Λ(f),α = Λ(f), ev(u) for every f L m ( ). Since R(Λ) is dense in CH 0 ( ) (by corollary 4.3), we conclude that α = ev(u). Remark 5.2. Note that the evaluation map is in fact an isomorphism of the Banach spaces BV 1 ( )[ BV1 ] and CH 0 ( ), according to the open mapping theorem. 6. Proof of the main theorem Theorem 6.1. Let F be a distribution in. The following conditions are equivalent: (i) there exists v C 0 ( ; ) such that Φ(v) =F ; (ii) F is a charge vanishing at infinity.

11 Distributional divergence 75 Proof. That (i) implies (ii) is proven by proposition 3.5. In order to prove that (ii) implies (i) we shall first show that R(Φ) is dense in CH 0 ( ), and then we will establish that R(Φ) is closed in CH 0 ( ) as an application of the closed range theorem. In order to show that R(Φ) is dense in CH 0 ( ), it suffices to prove the following, according to the Hahn Banach theorem. Every α CH 0 ( ) whose restriction to R(Φ) is zero vanishes identically. Assume α CH 0 ( ) and Φ(v),α = 0 for every v C 0 ( ; ). It follows from proposition 5.1 that α =ev(u) for some u BV 1 ( ). Since 0= Φ(v), ev(u) = v, d( u) for every v, we infer that u = 0, and in turn u = 0. Thus, α =ev(u) = 0 and the proof that R(Φ) is dense in CH 0 ( ) is complete. In order to show that R(Φ) is closed in CH 0 ( ), it suffices to show that R(Φ ) is closed in C 0 ( ; ), according to the closed range theorem. We first need to identify the adoint map Φ of Φ. Recall that CH 0 ( ) is identified with BV 1 ( ) through the evaluation map (see proposition 5.1), and that C 0 ( ; ) is identified with M( ; ). Given α CH 0 ( ), we find u BV 1 ( ) such that α =ev(u). For each v C 0 ( ; ), we have v, Φ (ev(u)) = Φ(v), ev(u) = u, Φ(v) = v, d( u). Thus, Φ ev =. Now let {α } be a sequence in CH 0 ( ) such that {Φ (α )} converges to some µ M( ; ). We ought to prove the existence of u BV 1 ( ) such that µ = u. Find a sequence {u } in BV 1 ( ) such that α =ev(u ). Observe that Φ (α ) M = (Φ ev)(u ) M = u M. Since {Φ (α )} is bounded, we infer that sup u M <. Then there exists a subsequence {u k } and u BV 1 ( ) such that u u k 0 according to proposition 2.6. In particular, for each v D( ; ), we have v, d( u) = u div v From this we infer that = lim k u k div v = lim k v, d( u k ). v, d( u) = v, dµ because µ is the limit of { u }. Since D( ; ) is dense in C 0 ( ; )we conclude that u = µ.

12 76 Th. De Pauw and M. Torres Corollary 6.2. For every f L m ( ), there exists v C 0 ( ; ) such that Λ(f) =Φ(v). 7. Another proof Here we provide an alternative approach based on the general theory developed in [3, 3]. Our space X =BV 1 ( ) is initially equipped with the locally convex linear topology T, which is the trace on BV 1 ( ) of the weak topology of L 1 ( ). We further consider the linearly stable family C [3, definition 3.1] consisting of those convex sets C := BV 1 ( ) {u: u M }, =1, 2,... The corresponding locally convex topology T C on BV 1 ( ) is described in [3, theorem 3.3]. We note that the bounded subsets of L 1 ( ) are weakly relatively compact (according to the Banach Alaoglu theorem [5, theorem 3.15], because L 1 ( )is reflexive) and that the restriction of the weak topology to such subsets is metrizable (because the dual of L 1 ( ) is separable [5, theorem 3.8(c)]). We infer from proposition 2.5 that the sets C defined above are weakly bounded. Thus, the C are T relatively compact, the restriction of T to C is sequential (in fact, metrizable) and, in turn, the C are T compact according to proposition 2.3. Next we infer from [3, proposition 3.8(1)] that a sequence {u k } in BV 1 ( ) converges to zero in the sense of T C if and only if it converges to zero in the sense of definition 2.2. Since the restriction of T to each C is sequential, as noted above, the proof of [3, proposition 3.8(3)] shows that CH 0 ( )=BV 1 ( )[T C ]. The strong topology on CH 0 ( ), i.e. the topology of uniform convergence on bounded subsets of BV 1 ( )[T C ], is exactly the normed topology considered in proposition 3.2 according to [3, proposition 3.8(2)]. The T-compactness of the C then implies that CH 0 ( ) = BV 1 ( ), via the evaluation map, according to [3, theorem 3.16]. In other words, proposition 5.1 is established in this abstract fashion. The proof of theorem 6.1 remains unchanged. Acknowledgements The authors are indebted to Philippe Bouafia and Washek F. Pfeffer for their useful comments and suggestions. References 1 J. Bourgain and H. Brezis. On the equation div Y = f and application to control of phases. J. Am. Math. Soc. 16 (2003), Th. De Pauw and W. F. Pfeffer. Distributions for which div v = F has a continuous solution. Commun. Pure Appl. Math. 61 (2008), Th. De Pauw, L. Moonens and W. F. Pfeffer. Charges in middle dimensions. J. Math. Pures Appl. 92 (2009), N. C. Phuc and M. Torres. Characterization of the existence and removable singularities of divergence-measure vector fields. Indiana Univ. Math. J. 57 (2008), W. Rudin. Functional analysis (New York: McGraw-Hill, 1973). 6 M.Šilhavý. Currents that can be represented as boundaries of locally bounded or continuous vector fields. (In preparation.) (Issued 25 February 2011 )