Iwaniec–Šverák conjecture on openness of finite-distortion mappings

Determine whether every nonconstant mapping of finite distortion in dimensions n≥3 belonging to W^{1,p}(Ω;ℝⁿ) with p≥n and having outer distortion coefficient K^O_f∈L^{n−1}(Ω) is necessarily an open mapping.

Background

The paper studies mappings of finite distortion as a weak-regularity framework for deformations in compressible magnetoelasticity. A central topological issue is whether critical integrability of the outer distortion coefficient guarantees openness of such mappings. The authors describe the classical Iwaniec–Šverák conjecture, note that it is resolved in two dimensions, and state that the problem remains largely unresolved in dimensions n≥3.

The paper proves a partial positive result: openness is established in the critical regime KO_f∈L{n−1} under the additional Ciarlet–Nečas condition. Thus, the unrestricted conjecture remains an explicitly identified open problem beyond the assumptions treated in the paper.

References

In its classical form, the conjecture asks whether every nonconstant mapping of finite distortion $$\bm f\in W{1,p}\left(\Omega;\mathbb Rn\right), \qquad p\ge n, $$ with outer distortion coefficient $$KO_{\bm f}\in L{n-1}(\Omega) $$ is necessarily open. For $n=2$, the conjecture has been confirmed (see ), while for dimensions n\ge3 the problem remains largely open.

— Existence of homeomorphic minimizers via mappings of finite distortion in compressible magnetoelasticity  (2608.13323 - Dutta et al., 13 Aug 2026) in Section 1, subsection “Motivation from finite distortion theory”