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Anisotropic fractional perimeters. (English) Zbl 1291.52013

For the Borel set \(E\subset \mathbb{R}^n\), an origin-symmetric body \(K\subset \mathbb{R}^n\) and \(0<s<1\), the anisotropic fractional \(s\)-perimeter of \(E\) with respect to \(K\) is defined as \(P_s(E,K)=\int_E\int_{E^c} \frac{1}{\|x-y\|^{n+s}_K} dxdy\), where \(\|\cdot\|_K\) denotes the norm with unit ball \(K\) and \(E^c\) denotes the complement of \(E\) in \(\mathbb{R}^n\).
In the paper under review, \((1-s)P_s(E,K)\) is shown to converge to the anisotropic perimeter of \(E\) with respect to the moment body of \(K\) as \(s \to 1-\) (Theorem 4), moreover, a corresponding result is established for anisotropic fractional \(s\)-seminorms on \(BV(\mathbb{R}^n)\) (Theorem 8). The author also establishes the asymptotic of \(sP_s(E,K)\) as \(s \to 0+\) (Theorem 5).
It should be noted two other results of this paper: the minimizers of the anisotropic fractional \(s\)-isoperimetric inequality with respect to \(K\) are shown to converge to the moment body of \(K\) as \(s \to 1-\) (Theorem 7), and anisotropic fractional Sobolev inequalities are established (Theorem 9).

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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