Search results for "VARIATION"
showing 10 items of 2124 documents
On the Minimal Solution of the Problem of Primitives
2000
Abstract We characterize the primitives of the minimal extension of the Lebesgue integral which also integrates the derivatives of differentiable functions (called the C -integral). Then we prove that each BV function is a multiplier for the C -integral and that the product of a derivative and a BV function is a derivative modulo a Lebesgue integrable function having arbitrarily small L 1 -norm.
The essential variation of a function and some convergence theorems
1996
ВВОДИтсь ОпРЕДЕлЕНИ Е ВАРИАцИИ ФУНкцИИ, пР И кОтОРОМ ФОРМУлА $$V(F,E) = \int_E {|\bar DF(x)} |dx$$ спРАВЕДлИВА Дль пРОИ жВОльНОИ ФУНкцИИF И пРОИжВОльНОгО ИжМЕР ИМОгО МНОжЕстВАE НА ОтРЕжкЕ пРьМОИ. В т ЕРМИНАх ЁтОИ ВАРИАцИ И пОлУЧЕНы тЕОРЕМы О пОЧлЕННОМ ДИФФЕРЕНцИРОВАНИИ п ОслЕДОВАтЕльНОстИ Ф УНкцИИ И тЕОРЕМы О пРЕДЕльНОМ пЕРЕхОДЕ пОД жНАкОМ И НтЕгРАлА ДАНжУА-пЕРР ОНА.
Notions of Dirichlet problem for functions of least gradient in metric measure spaces
2019
We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a (1, 1)-Poincaré inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of Juutinen and Mazón-Rossi–De León, solutions by considering the Dirichlet problem for p-harmonic functions, p>1, and letting p→1. Tools developed and used in this paper include the inner perimeter measure of a domain. Peer reviewed
Strongly measurable Kurzweil-Henstock type integrable functions and series
2008
We give necessary and sufficient conditions for the scalar Kurzweil-Henstock integrability and the Kurzweil-Henstock-Pettis integrability of functions $f:[1, infty) ightarrow X$ defined as $f=sum_{n=1}^infty x_n chi_{[n,n+1)}$. Also the variational Henstock integrability is considered
The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces
2010
Abstract We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincare inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is us…
Some remarks on nonsmooth critical point theory
2006
A general min-max principle established by Ghoussoub is extended to the case of functionals f which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function, when f satisfies a compactness condition weaker than the Palais-Smale one, i.e., the so-called Cerami condition. Moreover, an application to a class of elliptic variational-hemivariational inequalities in the resonant case is presented. © Springer Science+Business Media B.V. 2007.
The Choquet and Kellogg properties for the fine topology when $p=1$ in metric spaces
2017
In the setting of a complete metric space that is equipped with a doubling measure and supports a Poincar´e inequality, we prove the fine Kellogg property, the quasi-Lindel¨of principle, and the Choquet property for the fine topology in the case p = 1. Dans un contexte d’espace m´etrique complet muni d’une mesure doublante et supportant une in´egalit´e de Poincar´e, nous d´emontrons la propri´et´e fine de Kellogg, le quasi-principe de Lindel¨of, et la propri´et´e de Choquet pour la topologie fine dans le cas p = 1. peerReviewed
The annular decay property and capacity estimates for thin annuli
2016
We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted $\mathbf{R}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar\'e inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincar\'e inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted $\mathbf{R}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also character…
Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents
1984
Abstract In this paper we study the existence of nontrivial solutions for the boundary value problem { − Δ u − λ u − u | u | 2 ⁎ − 2 = 0 in Ω u = 0 on ∂ Ω when Ω⊂Rn is a bounded domain, n ⩾ 3, 2 ⁎ = 2 n ( n − 2 ) is the critical exponent for the Sobolev embedding H 0 1 ( Ω ) ⊂ L p ( Ω ) , λ is a real parameter. We prove that there is bifurcation from any eigenvalue λj of − Δ and we give an estimate of the left neighbourhoods ] λ j ⁎ , λj] of λj, j∈N, in which the bifurcation branch can be extended. Moreover we prove that, if λ ∈ ] λ j ⁎ , λj[, the number of nontrivial solutions is at least twice the multiplicity of λj. The same kind of results holds also when Ω is a compact Riemannian manif…
Some new results on integration for multifunction
2018
It has been proven in previous papers that each Henstock-Kurzweil-Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable.