1,720,987 research outputs found
Functions of bounded variation in Carnot-Carathéodory spaces
Full text linkWe study properties of functions with bounded variation in Carnot-Carathéodory spaces. In Chapter 2 we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative. In Chapter 3 we prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes all Heisenberg groups H^n with n ≥ 2. Some important tools for the proof are properties linking the horizontal derivatives of a real-valued function with bounded variation to its subgraph. In Chapter 4 we prove a compactness result for bounded sequences (u_j) of functions with
bounded variation in metric spaces (X, d_j) where the space X is fixed, but the metric may vary with j. We also provide an application to Carnot-Carathéodory spaces. The results of Chapter 4 are fundamental for the proofs of some facts of Chapter 2
Fine properties of functions with bounded variation in Carnot-Carathéodory spaces
No full textWe study properties of functions with bounded variation in Carnot-Carathéodory spaces. We prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives. Under an additional assumption on the space, called property R, we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative
Distributional Solutions of Burgers’ type Equations for Intrinsic Graphs in Carnot Groups of Step 2
Full text linkWe prove that in arbitrary Carnot groups G of step 2, with a splitting G = W · L with L one-dimensional, the intrinsic graph of a continuous function φ : U ⊆ W → L is C1H -regular precisely when φ satisfies, in the distributional sense, a Burgers’ type system D φ φ = ω, with a continuous ω. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. We notice that our results generalize previous works by Ambrosio-Serra Cassano-Vittone and Bigolin-Serra Cassano in the setting of Heisenberg groups. As a tool for the proof we show that a continuous distributional solution φ to a Burgers’ type system D φ φ = ω, with ω continuous, is actually a broad solution to D φ φ = ω. As a by-product of independent interest we obtain that all the continuous distributional solutions to D φ φ = ω, with ω continuous, enjoy 1/2-little H ̈older regularity along vertical directions
Lipschitz minimizers for a class of integral functionals under the bounded slope condition
Full text linkWe consider the functional∫Ωg(∇u+X∗) dL2nwheregis convex andX∗(x,y)=2(−y,x)and we study the minimizers in BV(Ω) of the associated Dirichlet problem. We prove that,under the bounded slope condition on the boundary datum, and suitable conditions ong,there exists a unique minimizer which is also Lipschitz continuous. The assumptions ongallow to consider both the case with superlinear growth and the one with linear growth.Moreover neither uniform ellipticity nor smoothness ofgare assumed
SBV functions in Carnot–Carathéodory spaces
No full textWe introduce the space SBV_X of special functions with bounded X-variation in Carnot–Carathéodory spaces and study its main properties. Our main outcome is an approximation result, with respect to the BV_X topology, for SBV_X functions
A compactness result for BV functions in metric spaces
Full text linkWe prove a compactness result for bounded sequences (u_j) of functions with bounded variation in metric spaces (X,d_j) where the space X is fixed but the metric may vary with j. We also provide an application to Carnot–Carathéodory spaces
Local minimizers and gamma-convergence for nonlocal perimeters in Carnot groups
No full textWe prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi’s rectifiability theorem holds, we provide a lower bound for the Γ-liminf of the rescaled energy in terms of the horizontal perimeter
Rank-one theorem and subgraphs of BV functions in Carnot groups
Full text linkWe prove a rank-one theorem à la G. Alberti for the derivatives of vector-valued maps with bounded variation in a class of Carnot groups that includes Heisenberg groups H^n for n ≥ 2. The main tools are properties relating the horizontal derivatives of a real-valued function with bounded variation and its subgraph
A Rectifiability Result for Finite-Perimeter Sets in Carnot Groups
Full text linkIn the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Specifically, we consider subsets Γ that, in a way similar to intrinsic Lipschitz graphs, have a cone property: there exists an open dilation-invariant subset C whose translations by elements in Γ do not intersect Γ . However, a priori the cone C may not have any horizontal directions in its interior. In every Carnot group, we prove that the reduced boundary of every finite-perimeter subset can be covered by countably many subsets that have such a cone property. The cones are related to the semigroups generated by the horizontal half-spaces determined by the normal directions. We further study the case when one can find horizontal directions in the interior of the cones, in which case we infer that finite-perimeter subsets are countably rectifiable with respect to intrinsic Lipschitz graphs. A sufficient condition for this to hold is the existence of a horizontal one-parameter subgroup that is not an abnormal curve. As an application, we verify that this property holds in every filiform group, of either first or second kind
La Gioconda. O monumento : monologo di Barnaba / Ponchielli, comp. Don Sebastiano. La notte è serena : barcarola / Donizetti, comp ; Ferrucio Corradetti, BAR ; con accomp. a grande orchestra
No full textTitre uniforme : Ponchielli, Amilcare (1834-1886). Compositeur. [La Gioconda]. ExtraitTitre uniforme : Donizetti, Gaetano (1797-1848). Compositeur. [Dom Sébastien, roi de Portugal]Comprend : La Gioconda / Ponchielli, comp ; Ferrucio Corradetti, BAR ; con accomp. a grande orchestra ; Don Sebastiano / Donizetti, comp ; Ferrucio Corradetti, BAR ; con accomp. a grande orchestraEnregistrement : (Italie) Milan, 18-11-1907Contient une table des matière
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