Diffeomorphic approximation of $W^{1,1}$ planar Sobolev homeomorphisms

Abstract

Let Ω ⊆ R 2 be a domain and let f ∈ W 1,1 (Ω, R 2 ) be a homeomorphism (between Ω and f (Ω)). Then there exists a sequence of smooth diffeomorphisms f k converging to f in W 1,1 (Ω, R 2 ) and uniformly.2000 Mathematics Subject Classification. 46E35.

A Measure and Orientation Preserving Homeomorphism with Approximate Jacobian Equal −1 Almost Everywhere

“…It easily follows that this homeomorphism cannot be approximated by smooth diffeomorphisms in the Sobolev norm (see [22,Corollary 1.2]). This is in contract with the case n = 2 where every Sobolev homeomorphism can be approximated by smooth diffeomorphisms in the Sobolev norm, see [21,23]. The case n = 3 remains open.…”

Residually many BV homeomorphisms map a null set onto a set of full measure

“…W 1,1 homeomorphisms. In [4] the authors prove that it is possible to approximate a W 1,1 homeomorphism uniformly and in the W 1,1 norm by piecewise affine homeomorphisms. Moreover in Proposition 3.1 (1), it is equivalent to consider the variation V ar(φ, Q) or the energy E(φ), hence if one considers the metric space (Y, d Y ) as defined in the previous subsection, for p = 1, it is not difficult to adapt the arguments presented in Section 4, to prove that the set of W 1,1 homeomorphisms of Q onto itself mapping a set of measure smaller than 1/n onto a set of measure larger than 1 − 1/n are open and dense.…”

“…∞ < ε/4 and, together with property (3), this implies that d X (f, f ε ) < ε. Moreover properties (1), (2) and (4) imply that f ε ∈ A n .…”

On the piecewise approximation of bi-Lipschitz curves

“…This would be rather important for applications in the context of the non-linear elasticity. This problem and its partial solutions have an interesting history, one can see for instance the papers [2,5,3,4] to have an overview of what is now known.…”