Diffeomorphic deformation feasibility research toolkit

Family 368: The three-dimensional Ball–Evans approximation problem. First source-informed review, 9 October 2026 Australia/Brisbane.

Problem and potential new use

Registration and mesh researchers can study how Sobolev homeomorphisms in three dimensions can be approximated by smooth diffeomorphisms while retaining the same target domain.

Applicability and commercial boundary

Strong W1p existence of an approximating sequence does not supply a practical algorithm, finite-stage approximation budget or an automatic fold-free guarantee for a computed mesh. The initial map must already be a homeomorphism.

Initial business decision

Conditional research. Commercial demand and profitability remain hypotheses. Research usefulness is not evidence of a buyer budget.

Next verification action

Read the construction quantitatively and test an explicit map with certified injectivity before considering a registration backend.

Evidence scope

The catalog statement and main-paper abstract or introductory theorem passages were individually reviewed. Available family-level scope documents were inspected, with attention to version differences and exclusions. This is a first application triage, not a full proof audit or independent Lean verification. Family 376 has nine paper abstracts inspected; selected scopes are not assumed to cover all nine papers in full. Source revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. See source metadata.

Pinned sources

• Strong diffeomorphic approximation in three dimensions for 1≤p≤2
• Strong diffeomorphic approximation in three dimensions for p>2