Mappings of generalized finite distortion and continuity

Abstract

We study continuity properties of Sobolev mappings𝑓∈𝑊1,𝑛loc(Ω,ℝ𝑛),𝑛⩾2, that satisfy the following generalized finite distortion inequality||𝐷𝑓(𝑥)||𝑛⩽𝐾(𝑥)𝐽𝑓(𝑥) + Σ(𝑥)for almost every𝑥∈ℝ𝑛.Here𝐾∶ Ω→[1,∞)andΣ∶ Ω→[0,∞)are measurable functions. Note that whenΣ≡0, we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion𝐾∈𝐿∞(Ω), where a sharp condition for continuity is thatΣis in the Zygmund spaceΣlog𝜇(𝑒 + Σ) ∈ 𝐿1loc(Ω)for some𝜇>𝑛−1.We also show that one can slightly relax the boundedness assumption on𝐾to an exponential class exp(𝜆𝐾) ∈𝐿1loc(Ω)with𝜆>𝑛+1, and still obtain continuous solutions when Σlog𝜇(𝑒 + Σ) ∈ 𝐿1loc(Ω)with𝜇>𝜆. On the other hand, for all𝑝,𝑞 ∈ [1,∞] with 𝑝−1+𝑞−1=1, we construct a discontinuous solution with 𝐾∈𝐿𝑝loc(Ω)andΣ∕𝐾 ∈ 𝐿𝑞loc(Ω), including an example withΣ∈𝐿∞loc(Ω)and𝐾∈𝐿1loc(Ω).