Search results for "Lebesgue"
showing 10 items of 53 documents
A new result on impulsive differential equations involving non-absolutely convergent integrals
2009
AbstractIn this paper we obtain, as an application of a Darbo-type theorem, global solutions for differential equations with impulse effects, under the assumption that the function on the right-hand side is integrable in the Henstock sense. We thus generalize several previously given results in literature, for ordinary or impulsive equations.
HENSTOCK INTEGRAL AND DINI-RIEMANN THEOREM
2009
In [5] an analogue of the classical Dini-Riemann theorem related to non-absolutely convergent series of real number is obtained for the Lebesgue improper integral. Here we are extending it to the case of the Henstock integral.
Everywhere differentiability of viscosity solutions to a class of Aronsson's equations
2017
For any open set $\Omega\subset\mathbb R^n$ and $n\ge 2$, we establish everywhere differentiability of viscosity solutions to the Aronsson equation $$ =0 \quad \rm in\ \ \Omega, $$ where $H$ is given by $$H(x,\,p)==\sum_{i,\,j=1}^na^{ij}(x)p_i p_j,\ x\in\Omega, \ p\in\mathbb R^n, $$ and $A=(a^{ij}(x))\in C^{1,1}(\bar\Omega,\mathbb R^{n\times n})$ is uniformly elliptic. This extends an earlier theorem by Evans and Smart \cite{es11a} on infinity harmonic functions.
A Riemann-Type Integral on a Measure Space
2005
In a compact Hausdorff measure space we define an integral by partitions of the unity and prove that it is nonabsolutely convergent.
Failure of the local-to-global property for CD(K,N) spaces
2016
Given any K and N we show that there exists a compact geodesic metric measure space satisfying locally the CD(0,4) condition but failing CD(K,N) globally. The space with this property is a suitable non convex subset of R^2 equipped with the l^\infty-norm and the Lebesgue measure. Combining many such spaces gives a (non compact) complete geodesic metric measure space satisfying CD(0,4) locally but failing CD(K,N) globally for every K and N.
Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian
2014
Abstract We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype ∂ t u = − ∑ i = 1 m X i ⁎ ( | X u | p − 2 X i u ) where p ⩾ 2 , X = ( X 1 , … , X m ) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and X i ⁎ denotes the adjoint of X i with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M ; (ii) a doubling condition for the μ-measure of d-metric balls; and (iii) the validity of a Poincare inequality involving X and μ. Our results extend the recent work in [16] , [36] , to a more general setting including the model cases of (1)…
The metric-valued Lebesgue differentiation theorem in measure spaces and its applications
2021
We prove a version of the Lebesgue Differentiation Theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon-Nikod\'{y}m property.
A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space
2020
We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.
Ein Kriterium f�r die Approximierbarkeit von Funktionen aus sobolewschen R�umen durch glatte Funktionen
1981
The present paper provides a necessary and sufficient criterion for an element of a Sobolev space W k p (Ω) to be approximated in the Sobolev norm by Ck(En)-smooth functions. Here Ω is a bounded open set of n-dimensional Euclidean space En with convex closure $$\bar \Omega$$ and boundary ∂Ω having n-dimensional Lebesgue measure zero. No further boundary regularity (such as e.g. the segment property) is required.Our main tools are the Hardy-Littlewood maximal functions and a slightly strengthened version of a well-known extension theorem of Whitney.This work was inspired by and is very close in spirit to the pertinent parts of Calderon-Zygmund [6].
Variational Henstock integrability of Banach space valued functions
2016
We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum \nolimits _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum \nolimits _{n=1}^{\infty }x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a…