Abstract

Let U be a Hausdorff locally convex topological vector subspace of D′(Rn)\mathcal{D}'(\mathbb{R}^n) verifying suitable structure conditions. A characterization of the sets K ⊆ U that can be described as K = {u ∈ U: −⟨u, Dφ⟩ ∈ C for every φ ∈ D(Rn)\mathcal{D}(\mathbb{R}^n) with φ ≥ 0, ∫Rnφ(x)dx\int_{\mathbb{R}^n} \varphi(x)dx = 1} for some closed convex subset C of ℝⁿ is proved. As corollaries characterizations of the sets K that can be described as K = {u ∈ Wloc1,p(Rn)W^{1,p}_{\mathrm{loc}}(\mathbb{R}^n): Du ∈ C for a.e. x in ℝⁿ} or K = {u ∈ BVloc(Rn)BV_{\mathrm{loc}}(\mathbb{R}^n): meas(A)⁻¹ ∫AdDu\int_A dDu ∈ C for every nonempty bounded open set A of ℝⁿ} for some closed convex subset C of ℝⁿ are obtained. Similar results for subsets K of D′(Rn)\mathcal{D}'(\mathbb{R}^n), S′\mathcal{S}', Llocp(Rn)L^p_{\mathrm{loc}}(\mathbb{R}^n), C0(Rn)C^0(\mathbb{R}^n) are also proved.

Author information

Contact details are reproduced from the original publication and may be historical.

Antonio Corbo Esposito

Dipartimento di Ingegneria Industriale, Facoltà di Ingegneria, Università di Cassino, via Di Biasio 43, 03043 Cassino, Italy.

Riccardo De Arcangelis

Dipartimento di Ingegneria dell’Informazione e Matematica Applicata, Università degli Studi di Salerno, via Salvador Allende, 84081 Baronissi, Italy

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Suggested citation

A. Corbo Esposito, R. De Arcangelis. “A Characterization of Sets of Functions and Distributions on ℝ^n Described by Constraints on the Gradient.” Journal of Convex Analysis 3 (1996), No. 2, 167–194. https://doi.org/10.68381/jca03012

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Published by Heldermann Verlag, 1996. Rights now held by Banach Press.