Search results for "Bounded variation"
showing 10 items of 25 documents
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
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…
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
Neumann p-Laplacian problems with a reaction term on metric spaces
2020
We use a variational approach to study existence and regularity of solutions for a Neumann p-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure and supporting a Poincare inequality. Trace theorems for functions with bounded variation are applied in the definition of the variational functional and minimizers are shown to satisfy De Giorgi type conditions.
Relaxation of certain integral functionals depending on strain and chemical composition
2012
We provide a relaxation result in $BV \times L^q$, $1\leq q < +\infty$ as a first step towards the analysis of thermochemical equilibria.
MR3058477 Reviewed Ereú, Thomás; Sánchez, José L.; Merentes, Nelson; Wróbel, Małgorzata Uniformly continuous set-valued composition operators in the …
2011
In this paper it is established a property of a composition operator between spaces of functions of bounded variation in the sense of Schramm. Let X and Y be two real normed spaces, C a convex cone in X and I a closed bounded interval of the real line. Moreover let cc(Y) be the family of all non-empty closed convex and compact subsets of Y. The authors study the Nemytskij (composition) operator (HF)(t)=h(t,F(t)), where F: I \rightarrow C and h: I\times C \rightarrow cc(Y) is a given set-valued function. They show that if the Nemytskij operator $H$ is uniformly continuous and maps the space \Phi BV (I;C) of functions (from I to C) of bounded \Phi-variation in the sense of Schramm into the sp…
MR2789279 Aziz, Wadie; Leiva, Hugo; Merentes, Nelson; Rzepka, Beata A representation theorem for φ-bounded variation of functions in the sense of Rie…
2012
The authors consider the class $V_\varphi^R (I^b_a)$ of functions $f:I^b_a =[a_1,b_1]\times [a_2,b_2]\subset \mathbb{R}^2 \to \mathbb{R}$ with bounded $\varphi$-total variation in the sense of Riesz, where $\varphi: [0,+ \infty) \to [0,+ \infty)$ is nondecreasing and continuous with $\varphi(0)=0$ and $\varphi(t) \to +\infty$ as $t \to +\infty$. If we assume that $\varphi$ is also such that $\lim_{t \to +\infty}\frac{\varphi(t)}{t}= +\infty$, then we obtain the main result. Precisely, the authors give a characterization of function of two variables defined on a rectangle $I^b_a$ belonging to $V_\varphi^R (I^b_a)$. Clearly, this result is a generalization of the Riesz Lemma.
Regularity of the Inverse of a Sobolev Homeomorphism
2011
We give necessary and sufficient conditions for the inverse ofa Sobolev homeomorphism to be a Sobolev homeomorphism and conditions under which the inverse is of bounded variation.
APPROXIMATION OF BANACH SPACE VALUED NON-ABSOLUTELY INTEGRABLE FUNCTIONS BY STEP FUNCTIONS
2008
AbstractThe approximation of Banach space valued non-absolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock–Kurzweil–Pettis and a Denjoy–Khintchine–Pettis integrable function can be only scalarly approximated in the Alexiewicz norm by a sequence of step functions. In case of Henstock–Kurzweil–Pettis and Denjoy–Khintchine–Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact.
Time-dependent weak rate of convergence for functions of generalized bounded variation
2016
Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation generated by a simple symmetric random walk with time steps $2T/n$ and space steps $\pm \sigma \sqrt{T/n}$ where $\sigma > 0$. For quite irregular terminal conditions $g$ (bounded variation on compact intervals, locally H\"older continuous) the rate of convergence of $u^n(t,x)$ to $u(t,x)$ is considered, and also the behavior of the error $u^n(t,x)-u(t,x)$ as $t$ tends to $T$