Interior regularity of planar Mumford–Shah minimizers

Source and verification

This article explains family 366 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

The Mumford–Shah energy balances a function’s smoothness against the length of its jump set. Family 366 claims that planar interior cracks have only three local forms: a regular arc, a crack tip, or a triple junction. The key proposed exclusion rules out isolated compact pieces in global blow-up limits.

🏷️ Preliminaries: a free discontinuity problem

Let $\Omega \subset {\mathbb{R}}^2$ be a bounded Lipschitz domain and let $g\in L^{\infty }(\Omega )$ represent observed data. An admissible pair consists of a relatively closed set $K$ of finite length and a function $u\in L^2(\Omega )$ with square-integrable weak gradient away from $K$. The energy on a region $V$ is

$$\begin{array}{c}{\mathcal{F}}_g(u,K;V)\\ ={\int }_{V\setminus K}|\nabla u{|}^2+{\mathcal{H}}^1(K\cap V)\\ +{\int }_V|u-g{|}^2.\end{array}$$

The first term penalizes variation inside regions, the second penalizes edges, and the third penalizes disagreement with data. Both $u$ and $K$ are unknown, hence the term free discontinuity.

An absolute minimizer permits arbitrary admissible changes compactly supported inside $V⋐\Omega$. It does not restrict competitors to preserve the crack’s topology. A reduced crack satisfies $K={\mathrm{supp}}_{\Omega }({\mathcal{H}}^1\upharpoonright K)$, removing zero-length decorations invisible to the energy.

🏷️ The claimed planar classification

Family 366, as stated in the manuscript

For a reduced absolute minimizer with bounded fidelity data, every interior crack point has a neighborhood in which $K$ is a single $C^{1,\alpha }$ arc, an arc ending at that point, or three such arcs meeting at $120^{\circ }$. Only finitely many global connected components of $K$ meet any relatively compact interior region. (OpenAI, 2026)

The finiteness statement concerns components of the entire set $K$, not components of $K\cap V$ or of the complement. No finiteness up to the outer boundary is asserted.

🏷️ Blow-up scaling and the crack-tip exponent

At $x_0\in K$, rescale space by $r$ and the function by $r^{1/2}$:

$$\begin{array}{c}{u}_r(y)=r^{-1/2}u(x_0+ry),\\ K_r=(K-x_0)/r.\end{array}$$

The Dirichlet energy and crack length in the rescaled unit disk both equal their original values in $B_r(x_0)$ divided by $r$. For a locally bounded function and bounded data, the original fidelity term is order $r^2$, so after this division it is order $r$. This explains why blow-up limits solve the zero-fidelity problem. Handling componentwise additive constants is part of the precise compactness theory.

A model crack tip has the form $u(r,\theta )=c\sqrt{r}\sin (\theta /2)$ on a slit plane. Differentiating in polar coordinates gives $|\nabla u|\asymp r^{-1/2}$. Therefore

$${\int }_0^{\delta }|\nabla u{|}^{p}rdr\asymp {\int }_0^{\delta }{r}^{1-p/2}dr,$$

which is finite precisely when $p<4$. Also $|\nabla u|>t$ restricts the point to a disk of radius comparable to $t^{-2}$, with area of order $t^{-4}$. This is weak $L^4$ behavior. The manuscript claims the corresponding local weak-$L^4$ bound for minimizers; strong $L^4$ would fail at the model crack tip.

🏷️ Excluding compact crack components

The source first reduces the stated finite-length formulation to a rectifiable, locally bounded minimizing pair. Established compactness and classification results then leave one obstruction: a global limiting crack with connected complement might have a bounded component.

The proposed argument isolates a compact portion $A$ at positive distance from the rest of the crack. It splits the gradient into a field carrying the jumps on $A$ and a harmonic background extending through $A$. Translating $A$ preserves its length. The interaction energy between these two fields is harmonic in the translation parameter.

A harmonic function with a local minimum is constant. Minimality is used to put the interaction in exactly that situation. Independent variations of jump values on a regular subarc then force the background potential to be affine. The available energy-growth estimate makes its gradient zero, and another value-variation identity eliminates the remaining field. Removing $A$ would then save positive length without paying Dirichlet energy, contradicting minimality.

This is where the novel compact-piece exclusion enters. Once the only blow-up sets are a line, half-line, or three rays, epsilon-regularity transfers their shape to the original minimizer. A finite cover by these connected model neighborhoods gives the stated finiteness of global components on compact interior regions.

🏷️ Key steps and proof outlines

Isolate a movable compact set before varying it

A bounded connected component of a closed crack set need not have positive distance from its other components. The source first enlarges it to a compact set $A$ such that $B=H\setminus A$ is closed and $\mathrm{dist}(A,B)>0$. On a large compact truncation, components are intersections of relatively open-and-closed neighborhoods. A finite selection of these neighborhoods separates the chosen component from the truncation boundary. Local finite length bounds ${\mathcal{H}}^1(A)$, and the support condition makes that length positive.

This preparation is what makes all sufficiently small translations $A+t$ admissible without crossing $B$. It may collect several connected components into $A$. When ${\mathbb{R}}^2\setminus H$ is connected, the larger complement ${\mathbb{R}}^2\setminus B$ remains connected: it contains that dense connected set and is contained in its closure.

A whole-plane projection separates the compact jump

Choose a cutoff $\chi$ equal to one near $A$ and supported away from $B$, and put $h=\chi v$. In the Hilbert space of square-integrable vector fields, project its gradient density onto the closure of smooth compactly supported gradients. Write the projection as $\nabla \varphi$, and set

$$F=h-\varphi ,p=\nabla h-\nabla \varphi ,f=v-F,G=\nabla f.$$

Orthogonality gives $\mathrm{div}p=0$ distributionally. Since $p$ is a gradient off $A$, its curl is supported on $A$. The function $f$ extends harmonically across $A$. Outside a large disk, finite Dirichlet energy excludes the logarithmic and growing terms in the harmonic Fourier expansion of $\varphi$. Normalize its constant to zero; then $F=O(|x{|}^{-1})$ and $p=O(|x{|}^{-2})$.

These decay rates justify the eventual comparison. The remaining field $G$ has only linear energy growth, ${\int }_{B_R}|G{|}^2\le C(1+R)$, so it need not lie in global $L^2$. On a dyadic annulus the interaction is nevertheless bounded by $O(R^{1/2})O(R^{-1})=O(R^{-1/2})$, a summable series.

Translation turns minimality into a harmonic minimum

Translate the compact field and crack together while holding $f$ and $B$ fixed. Also allow an independent compact jump variation with projected field $q$ and amplitude $\sigma$. A distant cutoff makes the comparison compactly supported. Its gluing cost tends to zero because the field decays and the original energy grows only linearly.

Define $I_p(t)=\int G\cdot p(\cdot -t)$ and similarly $I_q$. Translation preserves crack length and the self-energy of $p$, giving

$$\begin{array}{rl}0\le & 2(I_p(t)-I_p(0))\\ & +2\sigma (I_q(t)+⟨p,q⟩)+{\sigma }^2\Vert q{\Vert }_2^2.\end{array}$$

The source proves $I_p$ harmonic in the translation variable. Near the translated crack, use $\Delta p={\nabla }^{\perp }\mathrm{curl}p$ and the fact that $G$ is a smooth gradient there; away from the crack, differentiate its harmonic kernel under the absolutely convergent integral. Setting $\sigma =0$, minimality gives a local minimum at $t=0$. The harmonic minimum principle therefore makes $I_p$ constant.

For each fixed $t$, the remaining polynomial in $\sigma$ has zero constant term and is nonnegative for both signs of $\sigma$. Its linear coefficient must vanish:

$$I_q(t)=-⟨p,q⟩.$$

Thus every compact jump variation has translation-independent interaction, a much stronger conclusion than stationarity of the original crack alone.

Arbitrary jumps force harmonic rigidity

On a regular arc $\gamma (s)$, choose a variation whose trace difference is an arbitrary smooth compactly supported function $\beta (s)$. Integration by parts expresses its interaction as

$$I_q(t)=\pm \int \beta (s)n(s)\cdot \nabla f(\gamma (s)+t)|{\gamma }^{\prime }(s)|ds.$$

Since this is constant in $t$ for every $\beta$, a fixed directional derivative of $f$ is constant on a small disk. In two dimensions harmonicity then makes $f$ affine there, and unique continuation makes it affine throughout the connected complement of $B$. Linear energy growth forces its constant gradient to vanish. Taking $q=p$ in the interaction identity then gives $\Vert p{\Vert }_2^2=0$. The original function is constant, so deleting the positive-length set $A$ strictly lowers the energy, contradicting minimality.

The final classification also uses the imported planar compactness and model-classification results: exclusion of compact components leaves only the empty set, a line, a half-line, or a $120$-degree triod as generalized limits. Bilateral Hausdorff convergence places an original minimizer near one of these models at a sufficiently small scale. The cited epsilon-regularity theorem upgrades that closeness to an actual local $C^{1,\alpha }$ chart. Classification of limits and this quantitative upgrade are distinct steps. (OpenAI, 2026 s “Isolating a compact piece,” “Excluding an isolated compact piece,” and “Classification and interior regularity”)

See Also

• Family 367 uses classification of homogeneous blow-ups to locate free-boundary singularities.
• Family 377 studies a different gradient-regularity problem where a model singularity also constrains the possible Hölder exponent.

🏷️ Manuscripts and verification scope

• Interior regularity of planar Mumford–Shah minimizers — pinned manuscript and companion source.

No family-specific scope page was available at lean/docs/366.md in this snapshot. This records the catalogue lookup, not a claim that no related Lean code exists. No Lean build or axiom audit was performed.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Interior regularity of planar Mumford–Shah minimizers.