Locally minimal sets for conformal dimension.
Measurability of classes of Lipschitz manifolds with respect to Borel -algebra of Vietoris topology
Measure extension for a piecewise invariant map
Measures on bundles and Trundles of measures
Measures representable as -dimensional Hausdorff measures
Menger curvature and rectifiability.
Mesures de Hausdorff et théorie de Perron-Frobenius des matrices non-négatives
Nous étudions des sous-ensembles parfaits de dont la structure dépend d’une matrice primitive à coefficients entiers . La dimension de Hausdorff d’un tel ensemble “fractal” s’exprime en fonction de la valeur propre réelle maximale de sa matrice associée. Nous utilisons le théorème de Perron-Frobenius pour calculer la valeur exacte (qui est finie et non-nulle) de la mesure de Hausdorff de cet ensemble, et nous montrons à quelle condition (géométrique) cette valeur est maximale.
Mesures spéciales dans l’espace
Mesures sur le cercle et convexes du plan
Minimizing movements for dislocation dynamics with a mean curvature term
We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution...
Misura di Hausdorff e misure approssimanti
Misure approssimanti ed insiemi ad ampiezza costante
Multifractals and projections.
In this paper, we generalize the result of Hunt and Kaloshin [5] about the Lq-spectral dimensions of a measure and that of its projections. The results we obtain, allow to study an untreated case in their work and to find a relationship between the multifractal spectrum of a measure and that of its projections.
(n,2)-sets have full Hausdorff dimension.
We prove that a set containing translates of every 2-plane must have full Hausdorff dimension.
Necessary and sufficient conditions for the chain rule in and
We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space and in the space of functions of bounded variation.
Non-isotropic distance measures for lattice-generated sets.
We study distance measures for lattice-generated sets in Rd, d>=3, with respect to non-isotropic distances l-l.K, induced by smooth symmetric convex bodies K. An effective Fourier-analytic approach is developed to get sharp upper bounds for the second moment of the weighted distance measure.
Note on coarea formulae in the Heisenberg group.
We show a first nontrivial example of coarea formula for vector-valued Lipschitz maps defined on the three dimensional Heisenberg group. In this coarea formula, integration on level sets is performed with respect to the 2-dimensional spherical Hausdorff measure, built by the Carnot-Carathéodory distance. The standard jacobian is replaced by the so called horizontal jacobian, corresponding to the jacobian of the Pansu differential of the Lipschitz map. Joining previous results, we achieve all possible...
Numerical Approximations of the Relative Rearrangement: The piecewise linear case. Application to some Nonlocal Problems
We first prove an abstract result for a class of nonlocal problems using fixed point method. We apply this result to equations revelant from plasma physic problems. These equations contain terms like monotone or relative rearrangement of functions. So, we start the approximation study by using finite element to discretize this nonstandard quantities. We end the paper by giving a numerical resolution of a model containing those terms.
On a class of nonlinear variational inequalities: high concentration of the graph of weak solution via its fractional dimension and Minkowski content.
On a general coarea inequality and applications.