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Camille Labourie

Publications and source records attributed to Camille Labourie.

15 recordsLinked to original sources
arXiv2026

Good access to the crack and integrability of the full gradient for Griffith almost-minimizers in the plane

We show that, for almost-minimizers of the Griffith energy in the plane, the complement of the crack is locally covered by a finite number of John domains where the displacement satisfies H\"older-type estimates. As a consequence, we derive the integrability of the full gradient and the finiteness of the traces along the crack. In particular, we show that any Griffith almost-minimizer in dimension two locally belongs to the space SBV^2.

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arXiv2026

Uniform rectifiability of brittle fractures in linear elasticity

We prove the uniform rectifiability of brittle fractures in arbitrary dimension. The existing approach for the Mumford-Shah functional, which relies on separation-type properties of the singular set, faces serious obstacles in the Griffith setting due to the lack of coarea formula for the symmetric gradient. We present an alternative route to uniform rectifiability for free-discontinuity problems by proving that cracks have ``plenty of big projections''.

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arXiv2025

Hausdorff dimension of the singular set for Griffith almost-minimizers in the plane

We consider regularity of the crack set associated to a minimizer of the Griffith fracture energy, often used in modeling brittle materials. We show that the crack is uniformly rectifiable which in conjunction with our previous epsilon-regularity result allows us to prove that the singular set has dimension strictly less than $1$. This size estimate also applies to almost-minimizers. As a byproduct, we prove higher integrability for the gradient of local minimizers of the Griffith energy, providing a positive answer to the analog of De Giorgi's conjecture for the Mumford--Shah functional.

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arXiv2025

An existence theorem for sliding minimal sets

We prove an existence theorem for the sliding boundary variant of the Plateau problem for $2$-dimensional sets in $\mathbb{R}^n$. The simplest case of sufficient condition is when $n=3$ and the boundary $\Gamma$ is a finite disjoint union of smooth closed curves contained in the boundary of a convex body, but the main point of our sufficient condition is to prevent the limits in measure of a minimizing sequence to have singularities of type $\mathbb{Y}$ along $\Gamma$.

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arXiv2025

Finite number of traces for Mumford-Shah minimizers in dimension 2

In this short note, we answer a question raised by E. De Giorgi, showing that a Mumford-Shah minimizer in dimension 2 can only admit three maximum limit values as approaching the singular set. This result stems from tools developed in the early 2000's by G. David, A. Bonnet, and J.-C. L\'eger.

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arXiv2024

Optimal regularity for quasiminimal sets of codimension one in $\R^2$ and $\R^3$

Quasiminimal sets are sets for which a pertubation can decrease the area but only in a controlled manner. We prove that in dimensions $2$ and $3$, such sets separate a locally finite family of local John domains. Reciprocally, we show that this property is a sufficient for quasiminimality. In addition, we show that quasiminimal sets locally separate the space in two components, except at isolated points in $\R^2$ or out a of subset of dimension strictly less than $N-1$ in $\R^N$.

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arXiv2024

Strong existence for free discontinuity problems in linear elasticity

In this note we show Ahlfors-regularity for a large class of quasiminimizers of the Griffith functional. This allows us to prove that, for a range of free discontinuity problems in linear elasticity with anisotropic, cohesive, or heterogeneous behavior, minimizers have an essentially closed jump set and are thus strong minimizers. Our notion of quasiminimality is inspired by and generalizes previous notions in the literature for the Mumford-Shah functional, and comprises functions which locally close to the crack have at most a fixed percentage of excess crack relative to minimizers. As for the case of minimizers of the Griffith functional, our proof of Ahlfors-regularity relies on contradiction-compactness and an approximation result for GSBD functions, showing the robustness of this approach with respect to generalization of bulk and surface densities.

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arXiv2023

Uniform concentration property for Griffith almost-minimizers

We prove that a Hausdorff limit of Griffith almost-minimizers remains a Griffith almost-minimizer. For this purpose, we introduce a new approach to the uniform concentration property of Dal Maso, Morel and Solimini which does not rely on the coarea formula, non available for symmetric gradient. We then develop several applications, including a general procedure to obtain global minimizers via blow-up limits.

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arXiv2023

An epsilon-regularity result for Griffith almost-minimizers in the plane

We present regularity results for the crack set of a minimizer for the Griffith fracture energy, arising in the variational modeling of brittle materials. In the planar setting, we prove an epsilon-regularity theorem showing that the crack is locally a $C^{1,1/2}$ curve outside of a singular set of zero Hausdorff measure. The main novelty is that, in contrast to previous results, no topological constraints on the crack are required. The results also apply to almost-minimizers.

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arXiv2022

Epsilon-regularity for Griffith almost-minimizers in any dimension under a separating condition

In this paper we prove that if (u, K) is an almost-minimizer of the Griffith functional and K is $\epsilon$-close to a plane in some ball B $\subset$ R N while separating the ball B in two big parts, then K is C 1,$\alpha$ in a slightly smaller ball. Our result contains and generalizes the 2 dimensional result of [4], with a different and more sophisticate approach inspired by [23, 24], using also [20] in order to adapt a part of the argument to Griffith minimizers.

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arXiv2020

Solutions of the (free boundary) Reifenberg Plateau problem

We solve two variants of the Reifenberg problem for all coefficient groups. We carry out the direct method of the calculus of variation and search a solution as a weak limit of a minimizing sequence. This strategy has been introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi and allowed them to solve the Reifenberg problem. We use an analogous strategy which allows to take into account the free boundary. Moreover, we show that the Reifenberg class is closed under weak convergence without restriction on the coefficient group.

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arXiv2020

Weak limits of quasiminimizing sequences

We show that the weak limit of a quasiminimizing sequence is a quasiminimal set. This generalizes the notion of weak limit of a minimizing sequences introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi. This result is also analogous to the limiting theorem of David in local Hausdorff convergence. The proof is based on the construction of suitable deformations and is not limited to the ambient space $\mathbb{R}^n \setminus \Gamma$, where $\Gamma$ is the boundary. We deduce a direct method to solve various Plateau problems, even minimizing the intersection of competitors with the boundary. Furthermore, we propose a structure to build Federer--Fleming projections as well as a new estimate on the choice of the projection centers.

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