On the total variations for the inverse of a BV-homeomorphism

Abstract

Let R 2 be a domain. In 2007 Hencl, Koskela and Onninen proved that if f W onto ! 0 is a homeomorphism of bounded variation then so does its inverse map f 1 D .x; y/W 0 ! . In this paper we present a different proof giving precise formulae for the total variations of the coordinate functions of f 1 , that is,As an application, we prove that weak*-compactness in BV holds simultaneously for sequences of BV-homeomorphisms f j and their inverses f 1 j ; this symmetry result fails in the setting of bi-Sobolev mappings. We also deduce by the above formulae a slight generalization of a recent theorem of Iwaniec and Onninen on the existence a.e. of a right inverse of weak limit of W 1;2 -homeomorphisms. Extensions to higher dimension are given.

On distributional adjugate and derivative of the inverse

“…This improves the result of [9] where the equality of variations in (1.6) is shown for W 1,n−1 homeomorphism for n ≥ 3.…”

“…By results of [8,9,13] we know that for f ∈ W 1,n−1 we have not only f −1 ∈ BV but also the total variation of the inverse satisfies where adj A denotes the adjugate matrix to A, i.e. the matrix of (n − 1) × (n − 1) subdeterminants arranged in such a way that A adj A = I det A.…”

Approximation of planar BV homeomorphisms by diffeomorphisms

“…Let us now select a point a = (a 1 , a 2 ) such that a 1 ∈ A, a 2 ∈ B, and let us define the auxiliary BV functions ω(z) = M (z − a) + u(a), and ψ(z) = u(z) − ω(z). For every x ∈ A, y ∈ (−3r, 3r), then, since ψ(a) = 0 we have 4) and the obvious modification of the argument implies that the estimate holds true also for x ∈ (−3r, 3r), y ∈ B. Let now (x, y) be any point in Q(c, 2r); thanks to (4.3) we can select x − < x < x + and y − < y < y + such that x ± ∈ A, y ± ∈ B, and |x + − x − | < r/R, |y + − y − | < r/R.…”

“…Before stating our main result, a couple of comments are useful. First of all, it is known that the inverse of a BV homeomorphism is also BV ( [7], see also [4,Theorem 1.3]). And in fact, in our construction not only the sequence {u j } area strict converges to u, but also the sequence {u −1 j } area strict converges to u −1 .…”

Bisobolev mappings and homeomorphisms with finite distortion

“…Recall that, in general, the inverse of a homeomorphism f a W 1; nÀ1 loc ðW; W 0 Þ belongs to BV loc only [3]. For planar homeomorphisms, the explicit expression of the total variation of the components of the inverse is given in [4] (see also [5]).…”