A Characterization of Sets of Functions and Distributions on ℝⁿ Described by Constraints on the Gradient

Let U be a Hausdorff locally convex topological vector subspace of $\mathcal{D}'(\mathbb{R}^n)$ 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 ∈ $\mathcal{D}(\mathbb{R}^n)$ with φ ≥ 0, $\int_{\mathbb{R}^n} \varphi(x)dx$ = 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 ∈ $W^{1,p}_{\mathrm{loc}}(\mathbb{R}^n)$: Du ∈ C for a.e. x in ℝⁿ} or K = {u ∈ $BV_{\mathrm{loc}}(\mathbb{R}^n)$: meas(A)⁻¹ $\int_A dDu$ ∈ 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 $\mathcal{D}'(\mathbb{R}^n)$, $\mathcal{S}'$, $L^p_{\mathrm{loc}}(\mathbb{R}^n)$, $C^0(\mathbb{R}^n)$ are also proved.