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In this paper, we study the existence of limits at infinity along almost every infinite curve for the upper and lower approximate limits of bounded variation functions on complete unbounded metric measure spaces. We prove that if the…

In the setting of the Euclidean space equipped with an arbitrary Radon measure, we prove the equivalence between several notions of function of bounded variation present in the literature. We also study the relation between various…

We introduce a new capacity associated to a non negative function V. We apply this notion to the study of a necessary and sufficient condition to ensure the existence and uniqueness of a Schrodinger type equation with measure data and with…

In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality. We show that at almost every point $x$ outside the Cantor and jump…

We study fine potential theory and in particular partitions of unity in quasiopen sets in the case $p=1$. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique…

We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda~Jr. We show that these two notions coincide, if the measure is doubling and supports a…

In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality, and obtain $L^1$…

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 $(1,1)$-Poincar\'e inequality.…

In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of BV - functions. The space of the…

In this paper, we prove trace-type Poincar\'e and Sobolev inequalities for the space of functions of bounded $\mathbb{A}$-Variation

By introducing an intrinsic perimeter measure for intrinsic countably rectifiable sets, we prove a compactness result and a Poincar\'e inequality for special functions with bounded variation in equiregular Carnot-Carath\'eodory spaces which…

We prove an integral representation result for variational functionals in the space $BV^{\mathcal{B}}$ of functions with bounded $\mathcal{B}$-variation where $\mathcal{B}$ denotes a $k$-th order, $\mathbb{C}$-elliptic, linear homogeneous…

We present the foundations of the theory of functions of bounded variation and sets of finite perimeter in abstract Wiener spaces.

This paper studies the relations between extendability of different classes of Sobolev $W^{1,1}$ and $BV$ functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak…

We establish certain fine properties for functions of bounded $\mathscr A$-variation known in the classical $BV$ setting. Here, $\mathscr A$ is a $k$th order constant-coefficient homogeneous linear differential operator with a…

The present paper is intended to provide the basis for the study of weakly differentiable functions on rectifiable varifolds with locally bounded first variation. The concept proposed here is defined by means of integration by parts…

We develop a theory of bounded variation functions and Besov spaces in abstract Dirichlet spaces which unifies several known examples and applies to new situations, including fractals.

In the setting of a metric space equipped with a doubling measure supporting a $(1,1)$-Poincar\'e inequality, we study the problem of minimizing the BV-energy in a bounded domain $\Omega$ of functions bounded between two obstacle functions…

We prove a Poincar\'e, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature…

The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on…