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A characterization of convex calibrable sets in \(\mathbb R^N\). (English) Zbl 1108.35073
The authors extend a known characterization of calibrable sets in \({\mathbb R}^2\) in term of the mean curvature of the boundary to every dimension. Precisely they prove that a convex and bounded set \(C \subset {\mathbb R}^n\) of class \(C^{1,1}\) is calibrable, i.e. once defined \(\lambda_C:= P(C)/| C| \) where \(P(A)\) denotes the perimeter of \(A\) and \(| A| \) the \(n\)-dimensional volume of \(A\), there is a vector field \(z \in L^{\infty}(C, {\mathbb R}^n)\), \(\| z\| _{\infty} \leqslant 1\), such that \[ - \text{div} \, z = \lambda_C \text{ in } \;C\, , z \cdot \nu^C = -1 \text{ in } \;\partial C\, . \] if and only if (\({\mathbf H}_C\) denotes the mean curvature) \[ (n-1) \text{ ess sup}_{x \in \partial C} {\mathbf H}_C (x) \leq \lambda_C \, . \]
Moreover \(C\) is a solution of the minimum problem \[ \min_{X \subset C} P(X) - \lambda_C | X| \, . \] As a by-product they obtain that any set \(C\) as above contain a convex calibrate set \(K\) and for every volume \(V \in [| K| , | C| ]\) the solution of \[ \min_{X \subset C\, , | X| =V} P(X) \] is a convex set. The evolution of sets \(C = C_1 \cup \dots \cup C_k\), with \(C_j\) of class \(C^{1,1}\) convex sets satisfying \({\overline C}_i \cap {\overline C}_j = \emptyset\), by the minimizing total variation flow is also described.

MSC:
35J70 Degenerate elliptic equations
49J40 Variational inequalities
35K65 Degenerate parabolic equations
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