The Gaussian measure on algebraic varieties
We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety is dense in the Hilbert space , where dμ denotes the volume form of M and the Gaussian measure on M.
The Geometric Traveling Salesman Problem in the Heisenberg Group.
The Hausdorff lower semicontinuous envelope of the length in the plane
We study the Hausdorff lower semicontinuous envelope of the length in the plane. This envelope is taken with respect to the Hausdorff metric on the space of the continua. The resulting quantity appeared naturally as the rate function of a large deviation principle in a statistical mechanics context and seems to deserve further analysis. We provide basic simple results which parallel those available for the perimeter of Caccioppoli and De Giorgi.
The mean curvature measure
We assign a measure to an upper semicontinuous function which is subharmonic with respect to the mean curvature operator, so that it agrees with the mean curvature of its graph when the function is smooth. We prove that the measure is weakly continuous with respect to almost everywhere convergence. We also establish a sharp Harnack inequality for the minimal surface equation, which is crucial for our proof of the weak continuity. As an application we prove the existence of weak solutions to the...
The multifractal spectrum of discrete measures
The volume near the zeroes of a smooth function.
Two counterexamples concerning Hausdorff dimensions of projections
Two problems on doubling measures.
Doubling measures appear in relation to quasiconformal mappings of the unit disk of the complex plane onto itself. Each such map determines a homeomorphism of the unit circle on itself, and the problem arises, which mappings f can occur as boundary mappings?
Type-II singularities of two-convex immersed mean curvature flow
We show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular...
Typical Dimension of the Graph of Certain Functions.
Über einige Invarianzeigenschaften der Eulerschen Charakteristik
Un confronto tra densità
Une Formule Approximative de Dimension pour Certain Produits de Riesz.
Uniform rectifiability and singular sets
Upper estimates on self-perimeters of unit circles for gauges
Let M² denote a Minkowski plane, i.e., an affine plane whose metric is a gauge induced by a compact convex figure B which, as a unit circle of M², is not necessarily centered at the origin. Hence the self-perimeter of B has two values depending on the orientation of measuring it. We prove that this self-perimeter of B is bounded from above by the four-fold self-diameter of B. In addition, we derive a related non-trivial result on Minkowski planes whose unit circles are quadrangles.
Variations of additive functions
We study the relationship between derivates and variational measures of additive functions defined on families of figures or bounded sets of finite perimeter. Our results, valid in all dimensions, include a generalization of Ward’s theorem, a necessary and sufficient condition for derivability, and full descriptive definitions of certain conditionally convergent integrals.
Weak notions of jacobian determinant and relaxation
In this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely the distributional Jacobian and the relaxed total variation, which in general could be different. We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals.
Weak notions of Jacobian determinant and relaxation
In this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely the distributional Jacobian and the relaxed total variation, which in general could be different. We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals.
Weighted norm inequalities relating the and the area functions
Аппроксимационные и дифференциальные свойства измеримых множеств