Constructive derivation of the porosity threshold ε(n) and of the explicit higher-integrability constants for Mumford–Shah minimizers

De Philippis and Figalli (2014) proved De Giorgi's conjecture that the gradient of a local minimizer of the Mumford–Shah functional is higher integrable in every dimension: there exist dimensional constants $\bar{\gamma}(n)>1$ and $\bar{C}(n)$ with $\int_{B_{1/2}\setminus K}|\nabla u|^{2\bar{\gamma}}\le\bar{C}$, and their proof is direct, so the constants are 'potentially computable' with a single exception: the smallness threshold $\varepsilon(n)$ of the porosity statement (their Proposition 2.1(iv)), inherited from Ambrosio–Fusco–Pallara, Rigot and Maddalena–Solimini, is obtained by a compactness-by-contradiction argument and is therefore not effective. The authors remark that removing this compactness step 'seems likely', but they do not carry it out. The open problem is to give a constructive, compactness-free derivation of $\varepsilon(n)$, and to propagate it through the De Philippis–Figalli argument, so that $C_\varepsilon$, $\bar{\gamma}(n)$ and $\bar{C}(n)$ become explicitly computable functions of the dimension alone. These constants are not computed anywhere in the literature, and all known proofs of the porosity input proceed by contradiction.

Background

The Mumford–Shah functional was introduced in 1989 by Mumford and Shah as a variational model for image segmentation: one seeks a closed 'edge set' $K$ and a function $u$ that is smooth off $K$, penalizing both the Dirichlet energy of $u$ outside $K$ and the size of $K$. For an open set $\Omega\subset\mathbb{R}^n$, a relatively closed set $K\subset\Omega$ and $u\in W^{1,2}(\Omega\setminus K)$ (the Sobolev space of square-integrable functions with square-integrable first derivatives), the energy is

$MS(u,K)[\Omega]=\int_{\Omega\setminus K}|\nabla u|^2\,dx+\mathcal{H}^{n-1}(K\cap\Omega),$

where $\mathcal{H}^{n-1}$ is $(n-1)$-dimensional Hausdorff measure, which on a smooth surface coincides with its area. A pair $(u,K)$ is a local minimizer in $\Omega$ if $MS(u,K)[B]\le MS(v,H)[B]$ for every ball $B\Subset\Omega$ and every competitor $(v,H)$ agreeing with $(u,K)$ outside $B$; existence of minimizers is classical since the late 1980s. In 1991 De Giorgi formulated a series of conjectures on local minimizers, one of which predicts that $|\nabla u|^2$ is locally integrable to some power $\gamma>1$, i.e. $\nabla u\in L^p$ for some $p=2\gamma>2$ ('higher integrability'). The interest is twofold: Ambrosio–Fusco–Hutchinson showed that $|\nabla u|\in L^p$ for some $p>2$ forces the singular set $\Sigma$ (the points of $K$ at which $K$ fails to be locally a $C^1$ surface) to satisfy $\dim_{\mathcal{H}}\Sigma\le\max\{n-2,\,n-p/2\}<n-1$; and explicit planar examples (the crack-tip minimizer) show that no exponent $p\ge4$ can hold, so 'some $p>2$' is the natural threshold.

De Lellis and Focardi established the conjecture in the plane in 2013, using the precise description of minimal Caccioppoli partitions of $\mathbb{R}^2$ (partitions into sets of least total perimeter, a two-dimensional tool); De Philippis and Figalli (2014), the source of this problem, settled it in every dimension. The all-dimensional proof is direct: $u$ is harmonic off $K$ with energy $\le C_0\rho^{n-1}$ on balls, so the super-level sets $\{|\nabla u|^2\ge M^{h+1}\}$ are contained in $M^{-h}$-neighborhoods of $K$; the essential additional input is a porosity statement for $K$ (Proposition 2.1(iv) of the paper): there is a dimensional constant $\varepsilon(n)>0$ such that for every $\varepsilon\in(0,\varepsilon(n))$ there is $C_\varepsilon>0$ with the property that for all $x\in K\cap B_1$ and $\rho<1$ one can find $y\in B_{\rho/2}(x)\cap K$ such that, up to a translation by $y$ and a rotation, $K$ coincides inside $B_{2\rho/C_\varepsilon}(y)$ with the graph of a $C^{1,1/4}$ function $f$ with small excess $\|\nabla f\|_\infty+\rho^{1/4}\|\nabla f\|_{C^{1/4}}\le C_0\varepsilon$ and with $\sup_{B_{2\rho/C_\varepsilon}(y)}\rho|\nabla u|^2\le C_0\varepsilon$. 'Porosity' means exactly this: at every point and every scale, a definite portion of some ball around the point is a near-flat piece of graph with low energy, so the non-graph part of $K$ cannot fill space. Removing such good balls iteratively decays the $\mathcal{H}^{n-1}$-measure of the remaining part of $K$ geometrically, which yields a power-law bound on the super-level sets and hence Theorem 1.1: there are dimensional constants $\bar{C}>0$ and $\bar{\gamma}=\bar{\gamma}(n)>1$ with

$\int_{B_{1/2}\setminus K}|\nabla u|^{2\bar{\gamma}}\le\bar{C}$

for all local minimizers in $B_2$.

The porosity statement itself is older: it goes back to Ambrosio–Fusco–Pallara (1997), with closely related versions by Rigot (2000) and Maddalena–Solimini (2001), and its proof is indirect: assuming the statement fails along a sequence of minimizers with thresholds $\varepsilon_j\downarrow0$, one rescales, extracts a convergent subsequence (the compactness step), and contradicts the $\varepsilon$-regularity theorem (smallness of a suitable excess forces $K$ to be a graph) for the limit. An existence-by-contradiction argument of this kind yields no usable value of the threshold $\varepsilon(n)$, and this is the single place where the constants of De Philippis–Figalli stop being computable. Footnote 1 of the 2014 paper states the situation precisely: the constants $\bar{C}$ and $\bar{\gamma}$ can be expressed explicitly in terms of the dimension and the constants $C_0$ and $C_\varepsilon$ of Proposition 2.1; while $C_0$ is computable, $C_\varepsilon$ depends on the threshold $\varepsilon(n)$, which 'is obtained in [Ambrosio–Fusco–Pallara] using a compactness argument'. The authors add that 'it seems likely that the compactness step could be avoided arguing as in [Schoen–Simon]' — referring to the direct, effective regularity proof for rectifiable currents (area-minimizing surfaces in geometric measure theory) minimizing parametric elliptic functionals — 'but since this would not give any new insight to the problem, we do not investigate further this point.' Consequently, no value, bound, or finite recipe for $\varepsilon(n)$, and therefore none for $C_\varepsilon$, $\bar{\gamma}(n)$ or $\bar{C}$, is presently available, even though every other constant in the proof is effectively traceable.

Problem Statement

Give a constructive, compactness-free derivation of the smallness threshold in the porosity statement for Mumford–Shah minimizers, and thereby make the constants of De Philippis–Figalli's higher-integrability theorem explicitly computable. Concretely, the task has two linked parts, the first being the load-bearing input for the second: (i) Prove, without any step that obtains a constant by compactness or contradiction, that for every local minimizer $(u,K)$ of the Mumford–Shah energy in $B_2\subset\mathbb{R}^n$ there is an explicitly specified constant $\varepsilon_0(n)>0$ (a closed-form expression in the dimension $n$, or an explicitly described finite computation evaluating it for each $n$) such that for every $\varepsilon\in(0,\varepsilon_0(n))$ there is an explicit $C_\varepsilon>0$ with the following property of Proposition 2.1(iv) of De Philippis–Figalli: for all $x\in K\cap B_1$ and all $\rho<1$ there exist $y\in B_{\rho/2}(x)\cap K$, a unit vector $\bar\nu\in\mathbb{R}^n$, and a $C^{1,1/4}$ function $f:\bar\nu^\perp\to\mathbb{R}$, where $\bar\nu^\perp$ is the hyperplane orthogonal to $\bar\nu$, with $f(0)=0$, such that

$K\cap B_{2\rho/C_\varepsilon}(y)=\big(y+\{z+f(z)\bar\nu:\ z\in\bar\nu^\perp\}\big)\cap B_{2\rho/C_\varepsilon}(y),$

$\|\nabla f\|_\infty+\rho^{1/4}\|\nabla f\|_{C^{1/4}}\le C_0\varepsilon,\qquad\sup_{B_{2\rho/C_\varepsilon}(y)}\rho|\nabla u|^2\le C_0\varepsilon,$

where $C_0$ is the computable dimensional constant of Proposition 2.1(ii)–(iii); in words, up to the translation by $y$ and a rotation, $K$ is the graph of $f$ inside $B_{2\rho/C_\varepsilon}(y)$ with small excess and low energy. (ii) Propagate this effective input through the De Philippis–Figalli argument — the normalization $\varepsilon=\min\{\varepsilon_0(n)/C_0,\,1/(12C_0)\}$ of their Remark 2.3 and the constants of their Lemmas 3.1–3.3 through the proof of Theorem 1.1 — so as to produce explicit computable expressions, depending only on the dimension $n$ and elementary structural data of the functional, for an exponent $\bar{\gamma}(n)>1$ and a constant $\bar{C}(n)$ such that

$\int_{B_{1/2}\setminus K}|\nabla u|^{2\bar{\gamma}}\le\bar{C}$

for every local minimizer in $B_2$, in every dimension $n$. The derivation must be compactness-free, with every constant explicit: the constants (or an explicit finite algorithm that evaluates them) must be stated, and no constant may be left as an untracked output of an existence-by-compactness argument. The statement extends with trivial modifications, noted by the source authors, to the full Mumford–Shah energy with a fidelity term $\int\alpha|u-g|^2$ and surface coefficient $\beta$; the two parts above form one objective, since (i) is precisely the missing effective input of (ii).

References

1. G. De Philippis and A. Figalli, Higher Integrability for Minimizers of the Mumford–Shah Functional, Archive for Rational Mechanics and Analysis 213 (2014), 491–502. DOI: 10.1007/s00205-014-0729-x, https://doi.org/10.1007/s00205-014-0729-x (author manuscript arXiv:1303.1196, https://arxiv.org/abs/1303.1196)
2. L. Ambrosio, N. Fusco, and D. Pallara, Partial regularity of free discontinuity sets. II, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie IV 24 (1997), no. 1, 39–62. zbMATH: 1088624, https://zbmath.org/?q=an%3A1088624; open-access full text at https://eudml.org/doc/84255
3. S. Rigot, Big pieces of $C^{1,\alpha}$-graphs for minimizers of the Mumford-Shah functional, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie IV 29 (2000), no. 2, 329–349. zbMATH: 1529122, https://zbmath.org/?q=an%3A1529122; open-access full text at https://eudml.org/doc/84409
4. F. Maddalena and S. Solimini, Regularity properties of free discontinuity sets, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire 18 (2001), no. 6, 675–685. DOI: 10.1016/s0294-1449(01)00078-6, https://doi.org/10.1016/s0294-1449(01)00078-6
5. R. Schoen and L. Simon, A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals, Indiana University Mathematics Journal 31 (1982), no. 3, 415–434. DOI: 10.1512/iumj.1982.31.31035, https://doi.org/10.1512/iumj.1982.31.31035; zbMATH: 3816474, https://zbmath.org/?q=an%3A3816474
6. E. De Giorgi, Free discontinuity problems in calculus of variations, in: Frontiers in Pure and Applied Mathematics (a collection of papers dedicated to J.-L. Lions on the occasion of his 60th birthday), R. Dautray ed., North-Holland, Amsterdam, 1991, pp. 55–62. zbMATH: 16936, https://zbmath.org/?q=an%3A16936; MR: 1110593
7. D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems, Communications on Pure and Applied Mathematics 42 (1989), no. 5, 577–685. DOI: 10.1002/cpa.3160420503, https://doi.org/10.1002/cpa.3160420503
8. L. Ambrosio, N. Fusco, and J. E. Hutchinson, Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford-Shah functional, Calculus of Variations and Partial Differential Equations 16 (2003), no. 2, 187–215. DOI: 10.1007/s005260100148, https://doi.org/10.1007/s005260100148
9. C. De Lellis and M. Focardi, Higher integrability of the gradient for minimizers of the 2d Mumford–Shah energy, Journal de Mathématiques Pures et Appliquées (9) 100 (2013), no. 3, 391–409. DOI: 10.1016/j.matpur.2013.01.006, https://doi.org/10.1016/j.matpur.2013.01.006
10. M. Focardi, Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture, in: N. Fusco and A. Pratelli (eds.), Free Discontinuity Problems, Springer, 2016, pp. 1–68. DOI: 10.1007/978-88-7642-593-6_1, https://doi.org/10.1007/978-88-7642-593-6_1 (preprint arXiv:1610.03784, https://arxiv.org/abs/1610.03784)
11. S. Piontek and T. Schmidt, Higher integrability for the gradient of Mumford-Shah almost-minimizers, ESAIM: Control, Optimisation and Calculus of Variations 26 (2020), article 6. DOI: 10.1051/cocv/2019063, https://doi.org/10.1051/cocv/2019063
12. C. Labourie and E. Milakis, Higher integrability of the gradient for the thermal insulation problem, Interfaces and Free Boundaries 25 (2023), no. 2, 193–216. DOI: 10.4171/ifb/481, https://doi.org/10.4171/ifb/481 (preprint arXiv:2101.09692, https://arxiv.org/abs/2101.09692)
13. C. De Lellis and M. Focardi, The regularity theory for the Mumford–Shah functional on the plane, EMS Monographs in Mathematics, EMS Press, 2025. DOI: 10.4171/emm/13, https://doi.org/10.4171/emm/13 (preprint arXiv:2308.14660, https://arxiv.org/abs/2308.14660)
14. M. Friedrich, C. Labourie, and K. Stinson, Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane, preprint (2025). arXiv:2510.08670, https://arxiv.org/abs/2510.08670
15. G. De Philippis and A. Figalli, A note on the dimension of the singular set in free interface problems, Differential and Integral Equations 28 (2015), no. 5/6. DOI: 10.57262/die/1427744099, https://doi.org/10.57262/die/1427744099 (preprint arXiv:1405.7827, https://arxiv.org/abs/1405.7827)