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Publication 14-CNA-003
More Counterexamples to Regularity for Minimizers of the Average Distance Problem Xin Yang Lu Abstract: The average distance problem, in the penalized formulation, involves solving $$ {\rm min}\ {E_\mu^\lambda},\qquad E_\mu^\lambda(\cdot):=\int d(x,\cdot)d\mu(x)+\lambda{\cal H}^1(\cdot), {(1)} $$ among path-wise connected, closed sets with finite ${\cal H}^1$-measure, where $\mu$ is a given measure, $\lambda$ is a given parameter and $d(x,\cdot):=d_{\cal H}(\{x\},\cdot)$, with $d_{\cal H}$ denoting the Hausdorff distance. The average distance problem can be also considered among convex sets with perimeter and/or volume penalization, i.e. $$ {\rm min}\ {\cal E}(\mu,\lambda_1,\lambda_2),\hspace{.15in} {\cal E}(\mu,\lambda_1,\lambda_2)(\cdot):=\int d(x,\cdot)d\mu(x)+\lambda_1{\rm Per}(\cdot) +\lambda_2{\rm Vol}(\cdot), {(2)} $$ where $\mu$ is a given measure, $\lambda_1,\lambda_2\geq0$ are given parameters, and the unknown varies among convex sets. Very little is known about the regularity of minimizers. The aim of this paper is twofold: first, we provide a second approach in constructing minimizers of (1) which are not $C^1$ regular; second, using the same technique, we provide an example of minimizer of (2), under perimeter penalization only, whose border is not $C^1$ regular. Get the paper in its entirety as |
