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

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Abstract

We study functions of least gradient as well as related superminimizers and solutions of obstacle problems in complete metric spaces that are equipped with a doubling measure and support a Poincaré 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 for 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∨ 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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Acknowledgments

The research was funded by a grant from the Finnish Cultural Foundation. The author wishes to thank Nageswari Shanmugalingam for helping to derive the lower semicontinuity property of 1-superminimizers, and the anonymous referee for carefully reading the manuscript and giving helpful comments.

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  1. Department of Mathematics, Linköping University, Linköping, SE-581 83, Sweden

    Panu Lahti

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  1. Panu Lahti
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Lahti, P. Superminimizers and a weak Cartan property for p = 1 in metric spaces. JAMA 140, 55–87 (2020). https://doi.org/10.1007/s11854-020-0082-y

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  • DOI: https://doi.org/10.1007/s11854-020-0082-y

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