1,721,066 research outputs found
Sobolev-Poincaré inequalities for differential forms and currents
Full text linkIn this note we collect some results in R^n about (p,q) Poincaré and Sobolev inequalities for differential forms obtained in a joint research with Franchi and Pansu. In particular, we focus to the case p=1. From the geometric point of view, Poincaré and Sobolev inequalities for differential forms provide a quantitative formulation of the vanishing of the cohomology. As an application of the results obtained in the case p=1 we obtain Poincaré and Sobolev inequalities for Euclidean currents
Maxwell's equations in anisotropic media and carnot groups as variational limits
Full text linkLet G be a free Carnot group (i.e. a connected simply connected nilpotent stratified free Lie group) of step 2. In this paper, we prove that the variational functional generated by "intrinsic" Maxwell's equations in G is the γ-limit of a sequence of classical (i.e. Euclidean) variational functionals associated with strongly anisotropic dielectric permittivity and magnetic permeability in the Euclidean space
A GAMMA-CONVERGENCE APPROACH TO NON-PERIODIC HOMOGENIZATION OF STRONGLY ANISOTROPIC FUNCTIONALS
No full textComparing three possible hypoelliptic Laplacians on the 5-dimensional Cartan group via div-curl type estimates
Full text linkOn general Carnot groups, the definition of a possible hypoelliptic Hodge-Laplacian on forms using
the Rumin complex has been considered in (M. Rumin, “Differential geometry on C-C spaces and application
to the Novikov-Shubin numbers of nilpotent Lie groups,” C. R. Acad. Sci., Paris Sér. I Math., vol. 329, no. 11, pp.
985–990, 1999, M. Rumin, “Sub-Riemannian limit of the differential form spectrum of contactmanifolds,” Geom.
Funct. Anal., vol. 10, no. 2, pp. 407–452, 2000), where the author introduced a 0-order pseudodifferential operator
on forms. However, for questions regarding regularity for example, where one needs sharp estimates, this
0-order operator is not suitable. Up to now, there have only been very few attempts to define hypoelliptic Hodge-
Laplacians on forms that would allow for such sharp estimates. Indeed, this question is rather difficult to address
in full generality, the main issue being that the Rumin exterior differential dc is not homogeneous on arbitrary
Carnot groups. In this note, we consider the specific example of the free Carnot group of step 3 with 2 generators,
and we introduce three possible definitions of hypoelliptic Hodge-Laplacians.We compare how these three possible
Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain &
Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi
& Pansu for the Rumin complex on Heisenberg groups
Primitives of volume forms in Carnot groups
Full text linkIn the Euclidean space, it is known that a function f in L^2 of a ball B, with vanishing average, is the divergence of a vector field F in L^ 2 and the L^2-norm of F in the ball B is controlled by the L^2-norm of f in the same ball B (times a constant ). In this note, we prove a similar result in any Carnot group G for a vanishing average f in L^p, when 1\le p < Q, where Q is the so-called homogeneous dimension of G
A recursive basis for primitive forms in symplectic spaces and applications to Heisenberg groups
Full text linkThis paper is divided in two parts: in Section 2, we define recursively a privileged basis of the primitive forms in a symplectic space (V^2n , ω). Successively, in Section 3, we apply our construction in the setting of Heisenberg groups H^n , n ≥ 1, to write in coordinates the exterior differential of the so-called Rumin’s complex of differential forms in H^n
Continuous primitives for higher degree differential forms in Euclidean spaces, Heisenberg groups and applications
No full textIt is shown that higher degree exact differential forms on compact Riemannian n-manifolds possess continuous primitives whose uniform norm is controlled by their L^n norm. A contact sub-Riemannian analogue is proven, with differential forms replaced with Rumin differential forms
L1-Poincaré and Sobolev inequalities for differential forms in Euclidean spaces
Full text linkIn this paper, we prove Poincaré and Sobolev inequalities for differential forms in L1(Rn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007)
Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups
Full text linkThe -Sobolev inequality states that for compactly supported functions on the Euclidean -space,
the -norm of a compactly supported function
is controlled by the -norm of its gradient.
The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the -norm of a compactly supported differential
-form is controlled by the -norm of its exterior differential and its exterior codifferential (in special cases the
-norm must be replaced by the -Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms
A General Easy-to-use Expression for Uncertainty Evaluation in Residual Voltage Measurement
Full text linkThe paper addresses one of the new and most important issues arising when Low Power Voltage Transformers (LPVTs) are used in power network substations for evaluating, among others, the residual voltage measurement. Conversely to open-triangle inductive instrument transformers, the use of phase voltage transformers for measuring the residual voltage gets challenging due to the very high accuracy required for the three LPVTs. In the paper, a general expression to estimate the residual voltage measurement uncertainty, starting from the LPVTs accuracy, is presented. The effectiveness of the proposed approach is then confirmed with both Monte Carlo simulations and actual measurements on a general three-phase system
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