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 Math´ematiques et Physique, Universit´e Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium ([email protected]) Monica Torres Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA ([email protected]) (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 m vanishing at infinity if and only if F acts linearly on BVm/(m−1)(R ) (the space of functions in Lm/(m−1)(Rm) whose distributional gradient is a vector-valued measure) and satisfies the following continuity condition: F (uj ) converges to zero for each sequence {uj } such that the measure norms of ∇uj are uniformly bounded and m/(m−1) m uj 0 weakly in L (R ). 1. Introduction The equation ∆u = f ∈ Lm(Rm) need not have a solution u ∈ C1(Rm). In this paper we prove that, to each f ∈ Lm(Rm), there corresponds a continuous vector m m field, vanishing at infinity, v ∈ C0(R ; R ) such that div v = f weakly. In fact, we characterize those distributions F on Rm such that the equation div v = F admits m m a weak solution v ∈ C0(R ; R ). 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]. ∗ m In this paper m 2 and 1 := m/(m − 1). Let BV ∗ (R ) denote the subspace ∗ 1 of L1 (Rm) 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 m to be a linear functional F :BV1∗ (R ) → R such that F (uj) → 0 whenever 1∗ m uj 0 weakly in L (R ) and sup ∇uM < ∞. (1.1) j m We denote by CH0(R ) the space of charges vanishing at infinity and we note (see m proposition 3.2) that it is a closed subspace of the dual of BV1∗ (R ) (where the latter is equipped with its norm ∇uM). Examples of charges vanishing at infinity include the functions f ∈ Lm(Rm) (see proposition 3.4) and the distributional c 2011 The Royal Society of Edinburgh 65 66 Th. De Pauw and M. Torres m m divergence div v of v ∈ C0(R ; R ) (see proposition 3.5). Our main result thus consists in proving that the operator m m m C0(R ; R ) → CH0(R ): v → div v (1.2) is onto. This is done by applying the closed range theorem. For this purpose we m ∗ m identify CH0(R ) with BV1∗ (R ) via the evaluation map (see proposition 5.1). m m m This in turn relies on the fact that L (R ) is dense in CH0(R ) (see corollary 4.3, which is obtained by smoothing). Therefore, the adjoint of (1.2) is m m m BV1∗ (R ) →M(R ; R ): u →−∇u. The observation that this operator has a closed range follows from compactness in m BV1∗ (R ) (see proposition 2.6). m Charges vanishing at infinity happen to be the linear functionals on BV1∗ (R ) which are continuous with respect to a certain locally convex linear (sequential, non- m metrizable, non-barrelled) topology TC on BV1∗ (R ). In other words, there exists a m locally convex topology TC on BV1∗ (R ) such that a sequence uj → 0 in the sense of TC if and only if the sequence {uj} 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 m ∗ ∼ m this perspective the key identification CH0(R ) = BV1∗ (R ) is simply saying m that BV1∗ (R )[TC] is semireflexive; a property which follows from the compactness proposition 2.6. 2. Preliminaries A continuous vector field v : Rm → Rm is said to vanish at infinity if, for every ε>0, there exists a compact set K ⊂ Rm such that |v(x)| ε whenever x ∈ Rm \ K. m m These form a linear space denoted by C0(R ; R ), which is complete under the m m m norm v∞ := sup{|v(x)|: x ∈ R }. The linear subspace Cc(R ; R ) (respectively, D(Rm; Rm)) consisting of those vector fields having compact support (respectively, m m smooth vector fields having compact support) is dense in C0(R ; R ). Thus, each m m ∗ element of the dual, T ∈ C0(R ; R ) , is uniquely associated with some vector- valued measure µ ∈M(Rm; Rm) as follows: T (v)= v, dµ , Rm according to the Riesz–Markov representation theorem. Furthermore, m m µM = sup v, dµ : v ∈D(R ; R ) and v∞ 1 . Rm A vector-valued distribution T ∈D(Rm; Rm)∗ with the property that m m sup{T (v): v ∈D(R ; R ) and v∞ 1} < ∞ m m extends uniquely to an element of C0(R ; R ) and is therefore associated with a vector-valued measure as above. Distributional divergence 67 We recall some properties of convolution. Let 1 p<∞, u ∈ Lp(Rm) and ϕ ∈D(Rm). For each x ∈ Rm, we define (u ∗ ϕ)(x)= u(y)ϕ(x − y)dy. Rm It follows from Young’s inequality that u ∗ ϕ ∈ Lp(Rm) and u ∗ ϕLp uLp ϕL1 . (2.1) Furthermore, u ∗ ϕ ∈ C∞(Rm) and ∇(u ∗ ϕ)=u ∗∇ϕ. In the case when ϕ is even and f ∈ Lq(Rm) with p−1 + q−1 =1,wehave f(u ∗ ϕ)= u(f ∗ ϕ). Rm Rm m We fix an approximate identity on R , {ϕk} [5, (6.31)], and we infer that lim u − u ∗ ϕkLp =0. (2.2) k Henceforth we assume that m 2. We let the Sobolev conjugate exponent of 1 be m 1∗ := . m − 1 ∗ Note that L1 (Rm) is isometrically isomorphic to Lm(Rm)∗. We will recall the Gagliardo–Nirenberg–Sobolev inequality ∗ ∇ ϕ L1 κm ϕ L1 whenever ϕ ∈D(Rm). m 1∗ m Definition 2.1. We let BV1∗ (R ) denote the linear subspace of L (R ) con- sisting of those functions u whose distributional gradient ∇u is a vector-valued measure, i.e. m m ∇uM = sup u div v : v ∈D(R ; R ) and v∞ 1 < ∞. Rm | | ∗ ∇ ∗ Rm Readily u := u L1 + u M defines a norm on BV1 ( ), which makes it a Banach space. In view of proposition 2.5, we will use the equivalent norm ∗ ∇ u BV1 := u M. m Definition 2.2. Given a sequence {uj} in BV1∗ (R ), we write uj 0 whenever ∇ ∞ (i) supj uj M < , 1∗ m (ii) uj 0 weakly in L (R ). m 1∗ m Proposition 2.3. Let {uj} be a sequence in BV1∗ (R ), u ∈ L (R ), and assume 1∗ m that uj uweakly in L (R ). It follows that ∇uM lim inf ∇ujM. (2.3) j 68 Th. De Pauw and M. Torres m m m m Proof. Let v ∈D(R ; R ) with v∞ 1. Since div v ∈ L (R ) and u u ∗ j weakly in L1 (Rm) we have, from definition 2.1, u div v = lim uj div v lim inf ∇ujM Rm j Rm j and, taking the supremum over all such v, we conclude that ∇uM lim inf ∇ujM. j The following density result is basic. m Proposition 2.4. Let u ∈ BV1∗ (R ). The following hold: m m (i) for every ϕ ∈D(R ), u ∗ ϕ ∈ BV1∗ (R ) and ∇(u ∗ ϕ)L1 ∇uMϕL1 ; (ii) if {ϕk} is an approximate identity, then u − u ∗ ϕk 0 and lim ∇(u ∗ ϕk)L1 = ∇uM; k m (iii) there exists a sequence {uj} in D(R ) such that u − uj 0 as well as lim ∇ujL1 = ∇uM. j ∗ Proof. We note that (2.1) yields u ∗ ϕ ∈ L1 .Wehave |∇(u ∗ ϕ)|(x)dx = |ϕ ∗∇u|(x)dx Rm Rm = ϕ(x − y)d∇u(y) dx Rm Rm |ϕ(x − y)| d∇u(y)dx Rm Rm = |ϕ(x − y)| dx d∇u(y) Rm Rm = ∇uMϕL1 , (2.4) which shows proposition 2.4(i). Let {ϕk} be an approximate identity. From proposition 2.4(i), we obtain ∇(u ∗ ϕk)M = |∇(u ∗ ϕk)|(x)dx ∇uMϕkL1 = ∇uM. (2.5) Rm 1∗ m 1∗ m Since u ∗ ϕk → u in L (R ), then, in particular, u ∗ ϕk uweakly in L (R ); i.e. m m f[(u ∗ ϕk) − u] → 0 for every f ∈ L (R ). (2.6) Rm Distributional divergence 69 From (2.5) and (2.6) we obtain that u − u ∗ ϕk 0. Moreover, from (2.5) and the lower semicontinuity (2.3) we conclude that limk ∇(u ∗ ϕk)L1 = ∇uM, which shows that proposition 2.4(ii) holds. m In order to establish (iii), we choose a sequence {ψi} in D(R ) such that 1B(0,i) ψi 1B(0,2i) and sup ∇ψiLm < ∞. (2.7) i As usual, let {ϕk} be an approximate identity. Referring to proposition 2.4(ii) we inductively define a strictly increasing sequence of integers {kj} such that |∇ ∗ | ∇ 1 (u ϕkj ) u M + . Rm j For each j and i, we observe that |∇ ∗ | | ∇ ∗ | | ∗ ∇ | [(u ϕkj )ψi] ψi (u ϕkj ) + (u ϕkj ) ψi . | ∗ |1∗ ∈ 1 Rm For fixed j we infer from (2.7) and the relation u ϕkj L ( ) that | ∗ ∇ | | ∗ ∇ | lim sup (u ϕkj ) ψi = lim sup (u ϕkj ) ψi i Rm i B(0,i)c ∗ 1/1 1∗ | ∗ | ∇ m lim sup u ϕkj ψi L i B(0,i)c =0.