The three-dimensional Ball–Evans approximation problem at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The three-dimensional Ball–Evans approximation problem

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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If this game doesn't work on your computer, we can't help you. No Lean version yet. Some unformalized games could have issues!

expertly designed by an internal OpenAI model

Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	fluids, heat, waves	Levels:	2
Category:	Partial differential equations	Lean version:	not yet
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The three-dimensional Ball–Evans approximation problem. Resolves the three-dimensional Ball–Evans approximation problem: every $W^{1,p}$ homeomorphism between arbitrary bounded domains in ℝ³, for $1\le p\lt \infty$, is a strong $W^{1,p}$ limit of smooth diffeomorphisms onto the same target.

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

released 2026-09-24 | 3 theorems · 25 lemmas · 33 proofs · 32,364 words | PLAY LEVEL 1 » (pdf)

We resolve the three-dimensional Ball–Evans approximation problem for $1\le p\le2$. 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.

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

released 2026-09-24 | 3 theorems · 49 lemmas · 68 proofs · 71,298 words | PLAY LEVEL 2 » (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.

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