Title
Values of finite distortion: continuity
Abstract
Let Ω ⊂ ℝⁿ, let K ∈ Lᵃ(Ω) with K ≥ 1, and let Σ ∈ Lᵇ(Ω) with Σ ≥ 0. In 2022, with Doležalová and Onninen, we set out to investigate the continuity of Sobolev mappings f : Ω → ℝⁿ with Lⁿ-integrable weak derivatives that satisfy |Df(x)|ⁿ ≤ K(x) Jf(x) + Σ(x) at a.e. x ∈ Ω. We found planar counterexamples when 1/a + 1/b ≥ 1, but were only able to give a positive result under a strong exponential integrability assumption on K. However, with Onninen, recently in August 2025, we finally discovered an argument which proves the continuity of such maps when 1/a + 1/b < 1. Our proof uses a non-standard Sobolev inequality involving sizes of superlevel sets that we have so far not found from the existing literature. We expect this continuity result to open up a path to a theory of "values of finite distortion" that generalizes the theory of mappings of finite distortion, in the same way as our previously developed theory of quasiregular values generalizes the theory of quasiregular maps.
