Search results for "upper gradient"
showing 6 items of 6 documents
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…
Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces
2014
We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal maps.
Tensorization of quasi-Hilbertian Sobolev spaces
2022
The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $X\times Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, $W^{1,2}(X\times Y)=J^{1,2}(X,Y)$, thus settling the tensorization problem for Sobolev spaces in the case $p=2$, when $X$ and $Y$ are infinitesimally quasi-Hilbertian, i.e. the Sobolev space $W^{1,2}$ admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces $X,Y$ of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally for $p\in (1,\infty)$ we…
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.
Higher integrability and stability of (p,q)-quasiminimizers
2023
Using purely variational methods, we prove local and global higher integrability results for upper gradients of quasiminimizers of a $(p,q)$-Dirichlet integral with fixed boundary data, assuming it belongs to a slightly better Newtonian space. We also obtain a stability property with respect to the varying exponents $p$ and $q$. The setting is a doubling metric measure space supporting a Poincar\'e inequality.
Regularity properties for quasiminimizers of a $(p,q)$-Dirichlet integral
2021
Using a variational approach we study interior regularity for quasiminimizers of a $(p,q)$-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincar\'{e} inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally H\"{o}lder continuous and they satisfy Harnack inequality, the strong maximum principle, and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for H\"older continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we cons…