2025/09/24 by Ilmari Kangasniemi, Kangasniemi, Ilmari, Jani Onninen +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Contact Mechanics and Variational Inequalities #Analytic and geometric function theory
paper · pdf · doi:10.48550/arxiv.2509.20326
We prove the continuity of Sobolev functions φ∈ W1,nloc(Ω), Ω⊂ ℝn, that satisfy |∇ φ(x)|n ≤ K(x)(⟨ ∇ φ(x), ξ(x)⟩ + A(x)), where ξ∈ Llocn/(n-1)(Ω, ℝn) is weakly divergence-free, and K ∈ Lploc (Ω), A ∈ Lqloc (Ω) are non-negative with p-1+q-1<1. The result is applicable to a broad class of differential inequalities of null Lagrangian type. As our principal application, we obtain a sharp continuity theorem for f ∈ W1,nloc (Ω, ℝn) satisfying the distortion inequality with defect | Df(x)|n ≤ K(x)det Df(x) + Σ(x); this result is new even in the planar case, and closes a significant gap between existing methods and known counterexamples. The proof relies on an overlooked Sobolev-type inequality formulated in terms of measures of superlevel sets.