The total variation TV(u) of the Jacobian determinant of nonsmooth vector fields u has recently been studied in [2] [3]. We focus on the subclass u(x) = ϕ(x/|x|) of homogeneous extensions of smooth functions ϕ : ∂Bn → Rn. In the case n = 2, we explicitely compute TV(u) for some relevant examples exhibiting a gap with respect to the total variation |DetDu| of the distributional determinant. We then provide examples of functions with |DetDu| = 0 and TV(u) = +∞. We finally show that this gap phenomenon doesn’t occur if n ≥ 3.
Remarks on the total variation of the Jacobian / Mucci, Domenico. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - 13:2(2006), pp. 223-233. [10.1007/s00030-005-0037-0]
Remarks on the total variation of the Jacobian
MUCCI, Domenico
2006
Abstract
The total variation TV(u) of the Jacobian determinant of nonsmooth vector fields u has recently been studied in [2] [3]. We focus on the subclass u(x) = ϕ(x/|x|) of homogeneous extensions of smooth functions ϕ : ∂Bn → Rn. In the case n = 2, we explicitely compute TV(u) for some relevant examples exhibiting a gap with respect to the total variation |DetDu| of the distributional determinant. We then provide examples of functions with |DetDu| = 0 and TV(u) = +∞. We finally show that this gap phenomenon doesn’t occur if n ≥ 3.
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