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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 $(1,1)$-Poincare 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 [23, 29], solutions by considering the Dirichlet problem for $p$-harmonic functions, $p>1$, and letting $pto 1$. Tools developed and used in this paper include the inner perimeter measure of a domain.
For a given domain $Omega subset Bbb{R}^n$, we consider the variational problem of minimizing the $L^1$-norm of the gradient on $Omega$ of a function $u$ with prescribed continuous boundary values and satisfying a continuous lower obstacle condition
We describe the behavior of p-harmonic Greens functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincare inequality.
We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincare inequality. The notion of harmonicity is based on the Dirichlet form defined in
We prove a compactness result for bounded sequences $(u_j)_j$ of functions with bounded variation in metric spaces $(X,d_j)$ where the space $X$ is fixed but the metric may vary with $j$. We also provide an application to Carnot-Caratheodory spaces.
We develop the long-time analysis for gradient flow equations in metric spaces. In particular, we consider two notions of solutions for metric gradient flows, namely energy and generalized solutions. While the former concept coincides with the notion