The Ball-Evans approximation problem in three dimensions: strong approximation of Sobolev homeomorphisms by diffeomorphisms

Theorems: for nonempty bounded connected open $\Omega,\Lambda\subset\mathbb R^3$ and a homeomorphism $f\in W^{1,p}(\Omega;\mathbb R^3)$ onto $\Lambda$, there are $C^\infty$ diffeomorphisms $f_j:\Omega\to\Lambda$ with $\|f_j-f\|_{W^{1,p}}\to0$; proved for $1\le p\le2$ in one manuscript and for all finite $p>2$ in the other. The method first builds onto locally bi-Lipschitz approximants with small Sobolev error, then smooths them with fine pointwise control using three-dimensional PL topology (Moise, Bing, Hamilton). It does not give convergence of inverses, approximation by piecewise affine maps with inverse control, or anything in dimensions $n\ge4$ for $p\ge\lfloor n/2\rfloor$.

Theorems: for nonempty bounded connected open $\Omega,\Lambda\subset\mathbb R^3$ and a homeomorphism $f\in W^{1,p}(\Omega;\mathbb R^3)$ onto $\Lambda$, there are $C^\infty$ diffeomorphisms $f_j:\Omega\to\Lambda$ with $\|f_j-f\|_{W^{1,p}}\to0$; proved for $1\le p\le2$ in one manuscript and for all finite $p>2$ in the other. The method first builds onto locally bi-Lipschitz approximants with small Sobolev error, then smooths them with fine pointwise control using three-dimensional PL topology (Moise, Bing, Hamilton). It does not give convergence of inverses, approximation by piecewise affine maps with inverse control, or anything in dimensions $n\ge4$ for $p\ge\lfloor n/2\rfloor$.