The three-dimensional Ball–Evans approximation problem

What OpenAI claims

Resolves the three-dimensional Ball–Evans approximation problem: every W^1,p homeomorphism between arbitrary bounded domains in ℝ3, for 1 ≤ p < ∞, is a strong W^1,p limit of smooth diffeomorphisms onto the same target.

Papers (2)

1. Strong diffeomorphic approximation in three dimensions for 1≤p≤2

   24 September 2026 · Preprint, not peer reviewed · PDF

   We resolve the three-dimensional Ball–Evans approximation problem for 1 ≤ p ≤ 2. Every W^1,p homeomorphism between arbitrary bounded domains in ℝ3 is a strong W^1,p limit of smooth diffeomorphisms onto the same target.

2. Strong diffeomorphic approximation in three dimensions for p>2

   24 September 2026 · Preprint, not peer reviewed · PDF

   We resolve the three-dimensional Ball–Evans approximation problem for every finite p > 2. Every W^1,p homeomorphism between arbitrary bounded domains in ℝ3 can be approximated strongly in W^1,p by smooth diffeomorphisms onto the same target.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.