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On the Distributional Divergence of Vector Fields Vanishing at Infinity 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.
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