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arXiv · 1706.01873

Superminimizers and a weak Cartan property for $p=1$ in metric spaces

Panu Lahti
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Abstract

We study functions of least gradient as well as related superminimizers and solutions of obstacle problems in metric spaces that are equipped with a doubling measure and support a Poincar\'e inequality. We show a standard weak Harnack inequality and use it to prove semicontinuity properties of such functions. We also study some properties of the fine topology in the case $p=1$. Then we combine these theories to prove a weak Cartan property of superminimizers in the case $p=1$, as well as a strong version at points of nonzero capacity. Finally we employ the weak Cartan property to show that any topology that makes the upper representative $u^{\vee}$ of every $1$-superminimizer $u$ upper semicontinuous in open sets is stronger (in some cases, strictly) than the $1$-fine topology.

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Panu Lahti. 2017-06-06. Superminimizers and a weak Cartan property for $p=1$ in metric spaces. https://arxiv.org/abs/1706.01873

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