Bi-Sobolev mappings and elliptic equations in the plane

Abstract

Suppose that f = (u, v) is a homeomorphism in the plane of the Sobolev class W 1,1 loc such that its inverse is of the same Sobolev class. We prove that u and v have the same set of critical points. As an application we show that u and v are distributional solutions to the same non-trivial degenerate elliptic equation in divergence form. We study similar properties also in higher dimensions.

Anisotropic Sobolev homeomorphisms

“…This is a consequence of the following characteristic property of a bi-Sobolev map which was proved in [3], [13], [9]:…”

“…Another interesting property of a bi-Sobolev map f = (u, v) in the plane is that u and v have the same critical points ( [13], [17]). …”

“…Moreover, there are interesting interplay between the integrability of the distortions K f and K f −1 and the regularity of f and f −1 (see [13], Theorem 5). In our general context of W 1,1 -homeomorphisms there are different distortion functions which play a significant role (see Section 4).…”

“…Let Ω ⊆ R = (x, y) : Ω onto −→ Ω have been intensively studied (see [9], [11] -homeomorphism, we say that f is a bi-Sobolev map (see [13]). We recall that a W…”

“…The following Lemma is the main step towards the equality in the area formula (see Theorem 1.3 of [3] and also [13], where the case f ACL, i.e., absolutely continuous on lines, is treated). 1) × (−1, 1).…”