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).


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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