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

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Companion: Strong diffeomorphic approximation in three dimensions for p > 2 evidence for this event

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VibeMathed record: The Ball-Evans approximation problem in three dimensions: strong approximation of Sobolev homeomorphisms by diffeomorphisms evidence for this event

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 $p\ge\lfloor n/2\rfloor$. parent of this event

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

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