Rendiconti del Seminario Matematico della Università di Padova


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Volume 137, 2017, pp. 19–55
DOI: 10.4171/RSMUP/137-2

Example of minimizer of the average-distance problem with non closed set of corners

Xin Yang Lu[1]

(1) Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West, Burnside Hall, QC H3A 2K6, Montreal, Canada

The average-distance problem, in the penalized formulation, involves minimizing $$E_\mu^\lambda(\Sigma):=\int_{\mathbb R^d} \mathrm {inf}_{y\in\Sigma} |x-y|\mathrm d\mu(x)+\lambda\mathcal H^1(\Sigma),$$ among compact, connected sets $\Sigma$, where $\mathcal H^1$ denotes the 1-Hausdorff measure, $d\geq 2$, $\mu$ is a given measure and $\lambda$ a given parameter. Regularity of minimizers is a delicate problem. It is known that even if $\mu$ is absolutely continuous with respect to Lebesgue measure, $C^1$ regularity does not hold in general. An interesting question is whether the set of corners, i.e. points where $C^1$ regularity does not hold, is closed. The aim of this paper is to provide an example of minimizer whose set of corners is not closed, with reference measure $\mu$ absolutely continuous with respect to Lebesgue measure.

Keywords: Nonlocal variational problem, average-distance problem, regularity

Lu Xin Yang: Example of minimizer of the average-distance problem with non closed set of corners. Rend. Sem. Mat. Univ. Padova 137 (2017), 19-55. doi: 10.4171/RSMUP/137-2