An Epiperimetric Inequality for the Thin Obstacle problem

We prove an epiperimetric inequality for the thin obstacle problem, extending the pioneering results by Weiss on the classical obstacle problem (Invent. Math. 138 (1999), no. 1, 23-50). This inequality provides the means to study the rate of converge of the rescaled solutions to their limits, as well as the regularity properties of the free boundary

On the measure and the structure of the free boundary of the lower dimensional obstacle problem

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in $\mathbb{R}^{n+1}$ up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, we prove (i) local finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary, (ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary, (iii) classification of the frequencies up to a set of dimension at most (n-2) and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point

Phase field approximation of cohesive fracture models

We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=min\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings

Existence of minimizers for the 22d stationary Griffith fracture model

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove existence of strong minimizers, that is deformation fields which are continuously differentiable outside a closed jump set and which minimize the relevant energy. To this aim, we show that minimizers of the weak formulation of the problem, set in the function space $SBD^2$ and for which existence is well-known, are actually strong minimizers following the approach developed by De Giorgi, Carriero, and Leaci in the corresponding scalar setting of the Mumford-Shah problem

Improved estimate of the singular set of Dir-minimizing Q-valued functions via an abstract regularity result

In this note we prove an abstract version of a recent quantitative stratifcation priciple introduced by Cheeger and Naber (Invent. Math., 191 (2013), no. 2, 321-339; Comm. Pure Appl. Math., 66 (2013), no. 6, 965-990). Using this general regularity result paired with an $\varepsilon$-regularity theorem we provide a new estimate of the Minkowski dimension of the set of higher multiplicity points of a Dir-minimizing Q-valued function. The abstract priciple is applicable to several other problems: we recover recent results in the literature and we obtain also some improvements in more classical contexts.Comment: modified title; minor change