Smoothing three-dimensional deformations without folding or changing the target

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Family 368 · Partial differential equations
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Imagine deforming a solid object without allowing different material points to occupy the same location. A mathematical deformation should therefore be one-to-one. It may nevertheless be rough: its local stretching can exist only as a weak derivative, rather than as a continuously varying matrix.

Can every such rough deformation be approximated by perfectly smooth, invertible deformations, while keeping both the positions and the local stretching close?

This is the Ball–Evans approximation problem. Ordinary smoothing controls derivatives but can destroy injectivity. Purely topological approximation can preserve injectivity but say little about derivatives. Reconciling the two requirements is the difficulty.

Family 368 claims a complete positive answer in three dimensions for every finite exponent $1\le p<\infty$. M1 treats $1\le p\le2$; M2 treats $2<p<\infty$. Every approximating map has exactly the original target, even when the source and target domains have irregular boundaries.

2. Background and essential definitions

Topology and derivatives impose different conditions

A homeomorphism $f:\Omega\to\Lambda$ is a continuous bijection with continuous inverse. Here $\Omega,\Lambda\subset\mathbb R^3$ are nonempty bounded connected open sets. A homeomorphism preserves the topology of the object and prevents interpenetration, but need not possess classical derivatives everywhere.

A diffeomorphism is a smooth bijection whose inverse is also smooth. Smoothness and injectivity alone are insufficient: the one-dimensional map $x\mapsto x^3$ is smooth and bijective, but its inverse is not differentiable at zero. A diffeomorphism must have an invertible derivative at every interior point.

The Sobolev condition $f\in W^{1,p}(\Omega;\mathbb R^3)$ says that its three components and their weak first derivatives belong to $L^p$. Write $Df$ for the resulting matrix field. The larger $p$ is, the more strongly the norm penalizes large derivatives.

Strong approximation means

$$\int_\Omega\bigl(|f_j-f|^p+|Df_j-Df|^p\bigr)\,dx\longrightarrow0.$$

This controls both where material points move and how their local deformation matrices differ. Uniform convergence of positions alone would not imply it: functions can have very small oscillation amplitude and very large oscillating derivatives.

Fixed target and interior smoothness

The fixed-target requirement is

$$f_j(\Omega)=\Lambda\qquad\text{for every }j.$$

It is stronger than having target domains converge to $\Lambda$, or merely having maps whose images lie inside $\Lambda$. No part of the target may be omitted by an individual approximant.

Smoothness is required only on the open domains. If their boundaries are rough, the theorem does not claim that the maps extend smoothly to their closures. Nor does it prescribe agreement with a boundary trace that may not exist continuously.

A homeomorphism can have a singular derivative

The Jacobian determinant $\det Df$ measures oriented local volume change where the derivative exists. It can vanish on a large set for a Sobolev homeomorphism. Global injectivity does not force every infinitesimal linear approximation to be invertible.

This matters because a rank-one or rank-two derivative can carry substantial Sobolev energy even though its determinant is zero. For example,

$$A=\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}$$

has determinant zero but squared matrix norm one. Ignoring all points with singular derivatives would therefore discard energy that strong approximation must reproduce.

3. Development history and previous partial results

Ball formulated the approximation question in connection with nonlinear elasticity and explicitly attributed it to Evans. The motivation is to understand whether weakly differentiable admissible deformations can be replaced by smoother ones without violating the geometric constraints of the material.

The planar case was resolved earlier. Iwaniec, Kovalev and Onninen first treated $p=2$ using harmonic replacements and smoothing. Their general theorem covers $1<p<\infty$, using coordinatewise $p$-harmonic replacements. Hencl and Pratelli proved the endpoint $p=1$, with geometric extensions and an adapted grid. These results already preserve the target and do not assume Sobolev regularity of the inverse.

Stronger input assumptions lead to stronger approximation conclusions. Daneri and Pratelli treated bi-Lipschitz planar maps, meaning both directions have bounded local stretching globally. Pratelli obtained simultaneous forward and inverse Sobolev approximation for planar maps with both directions in $W^{1,1}$. These are distinct from merely requiring each approximant to have a smooth inverse.

Higher dimensions introduce real obstructions. Following Hencl and Vejnar, Campbell, Hencl and Tengvall constructed examples in dimensions $n\ge4$, for $1\le p<\lfloor n/2\rfloor$, with Jacobian determinants of both signs on sets of positive measure. Strong derivative approximation by diffeomorphisms cannot reproduce that sign pattern, because a diffeomorphism on a connected domain has one orientation.

Three dimensions behave differently. Hencl and Malý showed that a sense-preserving $W^{1,1}_{\mathrm{loc}}$ homeomorphism has nonnegative Jacobian almost everywhere. This removes the sign-changing obstruction but does not make the determinant strictly positive or construct an approximation.

A separate line of work smooths maps that are already piecewise affine. Campbell and Soudský obtained $C^1$ homeomorphic approximation in arbitrary dimension. Campbell, D’Onofrio and Vítek obtained diffeomorphic smoothing in dimensions three and four with derivative control. The remaining challenge for arbitrary Sobolev inputs is reaching such a tractable intermediate map while retaining strong derivative accuracy.

M1 and M2 claim to supply that missing approximation step in three dimensions, through two different constructions.

4. What the new manuscripts claim

Together the papers state the following: if $\Omega,\Lambda\subset\mathbb R^3$ are bounded domains and $f:\Omega\to\Lambda$ is a $W^{1,p}$ homeomorphism for a finite $p\ge1$, there exist $C^\infty$ diffeomorphisms

$$f_j:\Omega\longrightarrow\Lambda$$

that themselves belong to $W^{1,p}$ and converge strongly to $f$ in that space.

M1 covers the endpoint and critical range $1\le p\le2$; M2 covers $2<p<\infty$. Neither range theorem invokes the other. Their common smoothing component is supplied within each manuscript.

The theorem assumes no boundary regularity, no continuous extension of $f$ to the boundary, no Sobolev regularity of $f^{-1}$, and no nonvanishing-Jacobian condition. Orientation-reversing maps are included by reflecting the target before construction and undoing that reflection afterward.

A shared intermediate statement concerns locally bi-Lipschitz homeomorphisms $H$. Given any Sobolev error tolerance $\varepsilon>0$ and any positive continuous function $\xi$ on $\Omega$, it claims a smooth diffeomorphism $g$ onto the same target with

$$\|g-H\|_{W^{1,p}(\Omega)}<\varepsilon, \qquad |g(x)-H(x)|<\xi(x).$$

The variable tolerance can become arbitrarily small near the boundary. This fine positional control is useful for preserving the exact target while smoothing an intermediate map.

The main theorem concerns forward convergence. It does not claim $f_j^{-1}\to f^{-1}$ in a Sobolev norm, and it does not include $p=\infty$.

5. Why the advance matters

Importance: 5/5. If correct, the family resolves a central approximation problem at the interface of nonlinear elasticity, geometric mapping theory and three-dimensional topology across the entire finite-exponent range.

Strong approximation is the bridge between weak analytic models and smooth deformations. The theorem would show that, in three dimensions, retaining injectivity and the original target does not itself obstruct approximation in the forward Sobolev norm.

Its importance should not be confused with automatic convergence of every physical energy. An elastic energy may strongly penalize tiny determinants or depend on inverse derivatives. Those quantities need additional control beyond $Df_j\to Df$ in $L^p$.

6. The main idea

Separate topology from energy estimates

Both papers first construct a locally bi-Lipschitz homeomorphism close to $f$, then smooth it. Three-dimensional triangulation and piecewise-linear topology provide ways to modify maps while preserving injectivity and matching prescribed local pieces.

Those topological tools are qualitative. They do not give a useful bound for the derivative of an arbitrary extension. The construction therefore measures selected curves, surfaces or scalar coordinates directly, rather than treating the existence of an injective extension as an energy estimate.

It also separates points by the rank of $Df$. Full-rank data admit local affine or smooth invertible models. Rank-one and rank-two data require special treatment because their nonzero derivative energy cannot simply be thrown into an exceptional set. Rank-zero points contribute no derivative energy to retain.

M1: measure curves and surfaces before filling cells

For $p\le2$, the construction selects source curves and, at $p=2$, source surfaces. It compares their images with complementary-dimensional pieces of a target triangulation. Counting intersections supplies budgets for image lengths and areas, including multiplicity.

On retained rank-one regions, a curve with nearly minimal length and nearly affine endpoints must move mostly in the desired direction. This recovers its derivative rather than merely bounding its energy.

At rank two and $p=2$, the analogous tool is a sharp disk-parametrization estimate: a surface with nearly minimal area and controlled boundary parametrization can be reparametrized with nearly minimal Dirichlet energy and derivative close to the desired planar map. The boundary parametrization matters; area alone does not control how quickly a parametrized surface is traversed.

Remaining cells are filled using their already chosen boundaries. Radial compression has finite $p$-energy when $p$ is below the cell dimension. Thus it fills two-dimensional faces for $p<2$ and three-dimensional cells throughout this range. The endpoint $p=2$ needs the disk estimate for faces instead of that subcritical argument.

Additional critical-exponent tools select reverse rays and measure generic image surfaces, using known consequences of three-dimensional $W^{1,2}$ homeomorphism theory. Inverse bounded-variation regularity is used as a derived fact, not imposed as a hypothesis.

At $p=1$, ordinary uniform-convexity arguments are unavailable. M1 instead uses a squared length-defect estimate to recover derivative accuracy along the selected lines.

M2: retain two scalar coordinates

For $p>2$, the principal construction represents rank-deficient data through two scalar coordinates. On small retained patches these coordinates have prescribed affine first-order models. Their values determine mean gradients through boundary integrals; nearly sharp upper bounds on their gradient norms then force strong gradient closeness by uniform convexity.

The target is organized into a polyhedral two-dimensional complex outside protected full-dimensional regions. On its faces, two parameters describe radial position and position around the perimeter. In the source, families of surfaces realize their level sets.

These surfaces must be routed without creating unwanted intersections or multiplying derivative errors at every change of route. Reference paths constrain where the realized coordinate levels can lie: a substantially misplaced level would force too much path energy across many small spheres. The relevant lower bound grows under refinement precisely because $p>2$.

The construction controls the derivatives of the two parameters directly. It does not substitute the potentially enormous derivative of a topologically chosen chamber-filling map into the main energy estimate.

For $2<p<3$, a separate compression handles certain exceptional interiors. For $p\ge3$, the volume Lusin property—the relevant preservation of measure-zero sets—removes that particular obstruction. These distinctions explain why the higher-exponent proof is not just the low-exponent construction repeated with a larger power.

Smooth without losing the target

The intermediate locally bi-Lipschitz map is approximated by a locally finite piecewise-affine map and then smoothed near edges and vertices. Local errors are chosen summably, so the final estimate is global on $\Omega$, not merely valid on each compact subset.

To preserve the target, the positional error is kept smaller than a fixed fraction of the distance to its boundary. The resulting deformation stays inside $\Lambda$ and gives a proper homotopy to the original homeomorphism. Combined with positive local degree and global degree one, this yields bijectivity onto the whole target for each approximant.

Throughout the construction, finite geometric constants are fixed before the error regions they multiply are made small. This ordering is how qualitative topology and quantitative energy control are kept compatible.

7. Scope, limitations, and open questions

The combined claim concerns bounded domains in $\mathbb R^3$ and finite $p$. It does not resolve every higher-dimensional Ball–Evans problem; the sign-changing examples already prevent a universal positive theorem in some higher-dimensional ranges.

Smoothness is interior, with no claimed smooth boundary extension or exact prescribed boundary trace. Although every approximant has a smooth inverse, convergence of those inverses in a chosen Sobolev space requires separate assumptions and estimates.

Strong derivative convergence also does not control an arbitrary singular determinant penalty. For example, a function of $\det Df_j$ that diverges as the determinant approaches zero need not have convergent integrals merely because the matrices converge in $L^p$.

Quantitative distortion bounds, simultaneous approximation in both directions, determinant-sensitive energy convergence and analogous results on other geometric spaces remain separate questions.

8. Sources and further reading

The planar approximation papers linked above establish the main positive precedents; the Campbell–Hencl–Tengvall paper explains the higher-dimensional sign obstruction. The recent piecewise-affine smoothing paper isolates the final regularization stage from the more difficult approximation of arbitrary Sobolev input.

For background, Sobolev spaces, diffeomorphisms and topological degree provide complementary introductions to the analytic and topological language used here.

Source manuscripts

• M1. OpenAI, Strong diffeomorphic approximation in three dimensions for 1≤p≤2 (September 24, 2026).
• M2. OpenAI, Strong diffeomorphic approximation in three dimensions for p>2 (September 24, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.