On a Theorem of de Giorgi and Stampacchia.
On a variant of Korn’s inequality arising in statistical mechanics
We state and prove a Korn-like inequality for a vector field in a bounded open set of , satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case are...
On a variant of Korn's inequality arising in statistical mechanics
We state and prove a Korn-like inequality for a vector field in a bounded open set of , satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case...
On a variational property of integral functionals and related conjectures
The aim of this paper is to give the proofs of those results that in [4] were only announced, and, at the same time, to propose some possible developments, indicating some of the most significant open problems.
On a variational theory of image amodal completion
On an optimal shape design problem in conduction
In this paper we analyze a typical shape optimization problem in two-dimensional conductivity. We study relaxation for this problem itself. We also analyze the question of the approximation of this problem by the two-phase optimal design problems obtained when we fill out the holes that we want to design in the original problem by a very poor conductor, that we make to converge to zero.
On calibrations for Lawson’s cones
On computer aided shape optimization
On Conditions for Unrectifiability of a Metric Space
We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.
On convex sets that minimize the average distance
In this paper we study the compact and convex sets K ⊆ Ω ⊆ ℝ2that minimize∫ Ω dist ( x ,K ) d x + λ 1 Vol ( K ) + λ 2 Per ( K ) for some constantsλ1 and λ2, that could possibly be zero. We compute in particular the second order derivative of the functional and use it to exclude smooth points of positive curvature for the problem with volume constraint. The problem with perimeter constraint behaves differently since polygons are never minimizers. Finally using a purely geometrical argument from...
On diffused-interface models of shape memory alloys
On embedded minimal disks in convex bodies
On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities
Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed...
On integral currents with constant mean curvature
On lower semicontinuity of multiple integrals
We give a new short proof of the Morrey-Acerbi-Fusco-Marcellini Theorem on lower semicontinuity of the variational functional . The proofs are based on arguments from the theory of Young measures.
On mass transportation.
On minimal hypersurfaces of nonnegatively Ricci curved manifolds.
On minimal laminations of the torus
On minimal surfaces with free boundaries in given homotopy classes
On new isoperimetric inequalities and symmetrization.