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

Let U be a Hausdorff locally convex topological vector subspace of

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) D ( R n )

with φ ≥ 0,

\int_{\mathbb{R}^n} \varphi(x)dx ∫ R n φ ( x ) d x

= 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) W l o c 1 , p ( R n )

: Du ∈ C for a.e. x in ℝⁿ} or K = {u ∈

BV_{\mathrm{loc}}(\mathbb{R}^n) B V l o c ( R n )

: meas(A)⁻¹

\int_A dDu ∫ A d D u

∈ 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) D ′ ( R n )

,

\mathcal{S}' S ′

,

L^p_{\mathrm{loc}}(\mathbb{R}^n) L l o c p ( R n )

,

C^0(\mathbb{R}^n) C 0 ( R n )

are also proved.