R esum e. Dans cet article nous prouvons un nouveau r esultat d'existence pour une classe de probl emes d'optimisation de forme assez g en erale. Les ouverts que nous consid erons poss edent une contrainte de nature g eom etrique sur la normale int erieure. Ce travail est motiv e par la formulation variationnelle d'un probl eme a f r o n ti ere libre dont la solution poss ede cette propri et e g eom etrique.

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MSC (2010) 32A26, 32F10, 32F20, 32W05We construct a global solution with C k -estimates for the∂-equation on q-convex intersections.

In [12] Michel and Perotti have shown C k estimates for solutions to the ∂-equation on piecewise strictly pseudoconvex domains. Our aim in this paper is to prove similar estimates on q-convex wedges. Classification (1991):32F20, 32F10, 32F40 0 Introduction Definition 0.1. A collection (U, ρ 1 , . . . , ρ m ) will be called a C ( ≥ 2) qconfiguration in C n if U ⊂ C n is a convex domain, and ρ 1 , . . . , ρ m are real C functions on U satisfying the following conditions: Mathematics Subjectiii) for all λ 1 , . . . , λ m ≥ 0 with λ 1 + . . . + λ m = 1, the Levi form at z of the function λ 1 ρ 1 + . . . + λ m ρ m has at least q + 1 positive eigenvalues.In the present paper we will prove the following:Theorem 0.2. Let (U, ρ 1 , . . . , ρ m ) be a C q-configuration. Then for each ξ ∈ U with ρ 1 (ξ) = . . . = ρ m (ξ) = 0, there is a radius R > 0 such that on the so-called q-convex wedge W = {z ∈ U : ρ j (z) < 0 for j = 1, . . . , m} ∩ {z ∈ C n : |z − ξ| < R} there exist linear operators T r : C 0 0,r (W ) → C 0 0,r−1 (W ),

Abstract. Let M be a smooth q-concave CR submanifold of codimension k in C n . We solve locally the ∂ b -equation on M for (0, r)-forms, 0 ≤ r ≤ q−1 or n−k−q+1 ≤ r ≤ n−k, with sharp interior estimates in Hölder spaces. We prove the optimal regularity of the ∂ boperator on (0, q)-forms in the same spaces. We also obtain L p estimates at top degree. We get a jump theorem for (0, r)-forms (r ≤ q − 2 or r ≥ n − k − q + 1) which are CR on a smooth hypersurface of M . We prove some generalizations of the Hartogs-Bochner-Henkin extension theorem on 1-concave CR manifolds.In [7] we proved the following Theorem 0.1. Let M be a C 2+l -smooth q-concave CR generic submanifold of codimension k in C n . Let z 0 ∈ M and s ∈ N with s ≤ n. Then there exist an open neighborhood M 0 ⊆ M of z 0 and kernels R s,r (ζ, z) for r = 0, . . . , q − 1, n − k − q, . . . , n − k with the following properties:(ii) R s,r (ζ, z) is of bidegree (s, r) with respect to z and of bidegree (n − s, n − k − r − 1) with respect to ζ;(iv) there is a constant C > 0 such that for every ε > 0, we have(v) for every domain Ω ⋐ M 0 with piecewise C 1 boundary, if f is a C 1 1991 Mathematics Subject Classification: 32F20, 32F10, 32F40.