Related papers: On rough traces of BV functions

We characterize lower semicontinuity of integral functionals with respect to weak$^*$ convergence in $\mathrm{BV}$, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the…

In the setting of a metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we show that the total variation of functions of bounded variation is lower semicontinuous with respect to $L^1$-convergence in…

An accurate functional inequality for Div-BV positive symmetric tensors $A$ in a bounded domain $U\subset\mathbb{R}^n$ arises whenever the tangential part of the normal trace $\gamma_\nu A\sim A\vec\nu$ is a finite measure over $\partial…

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into $L^q$ spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also…

A generalized divergence theorem is established allowing for domains with inner boundaries. The normal trace of a rough integrand is not a Radon measure; rather, the boundary integral is expressed via a surface functional continuous with…

There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case where the Sobolev exponent p is less than one…

We introduce a family of pairings between a bounded divergence-measure vector field and a function $u$ of bounded variation, depending on the choice of the pointwise representative of $u$. We prove that these pairings inherit from the…

For \Omega a C^{2}-smooth domain, and a positive bounded continuous map a \in C(\Omega), we prove existence of a minimizer of the functional u \mapsto $\int_{\Omega} a|Du| over the space BV(\Omega) of functions of bounded variation with…

A full interpolation theory for Sobolev functions with smoothness between 0 and 1 and vanishing trace on a part of the boundary of an open set is established. Geometric assumptions are of mostly measure theoretic nature and reach beyond…

The identification between the complex plane and the Riemann sphere preserves holomorphic and harmonic functions and is a classical tool. In this paper we consider a similar mapping from an unbounded metric space $X$ to a bounded space and…

In the setting of a non-complete doubling metric measure space $(\Omega,d,\mu)$, we construct various bounded linear trace and extension operators for homogeneous and inhomogeneous Besov spaces $B^\alpha_{p,q}$. Equipping the boundary…

We characterize all (signed) measures in $BV_{\frac{n}{n-1}}(\mathbb{R}^n)^*$, where $BV_{\frac{n}{n-1}}(\mathbb{R}^n)$ is defined as the space of all functions $u$ in $L^{\frac{n}{n-1}}(\mathbb{R}^n)$ such that $Du$ is a finite…

In a Banach space $X$ endowed with a nondegenerate Gaussian measure, we consider Sobolev spaces of real functions defined in a sublevel set $O= \{x\in X:\;G(x) <0\}$ of a Sobolev nondegenerate function $G:X\mapsto \R$. We define the traces…

We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as…

We give upper bounds for the Poincar\'e and Logarithmic Sobolev constants for doubly weighted Brownian motion on manifolds with sticky reflecting boundary diffusion under curvature assumptions on the manifold and its boundary. We therefor…

In this paper we investigate Poincar\'e-type integral inequalities in the functional Musielak structure. We extend the ones already well known in Sobolev, Orlicz and variable exponent Sobolev spaces. We introduce conditions on the Musielak…

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with…

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established,…

In the paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We derive exact and easily computable constants for some…

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…