Topological equivalence of some variational problems involving distances

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April 2001, 7(2): 247-258. doi: 10.3934/dcds.2001.7.247

Topological equivalence of some variational problems involving distances

Giuseppe Buttazzo ^1, , Luigi De Pascale ^2, and Ilaria Fragalà ^3,

1.	Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, 56127 Pisa, Italy
2.	Dipartimento di Matematica, Via Buonarroti 2, 56127 Pisa, Italy
3.	Dipartimento di Matematica - Politecnico, Piazza Leonardo Da Vinci 32, 20133, Milano, Italy

Revised August 2000 Published January 2001

Abstract
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To every distance $d$ on a given open set $\Omega\subseteq\mathbb R^n$, we may associate several kinds of variational problems. We show that, on the class of all geodesic distances $d$ on $\Omega$ which are bounded from above and from below by fixed multiples of the Euclidean one, the uniform convergence on compact sets turns out to be equivalent to the $\Gamma$-convergence of each of the corresponding variational problems under consideration.

Keywords: geodesic distances, Gamma-convergence, length functionals..

Mathematics Subject Classification: 49J45, 53C6.

Citation: Giuseppe Buttazzo, Luigi De Pascale, Ilaria Fragalà. Topological equivalence of some variational problems involving distances. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 247-258. doi: 10.3934/dcds.2001.7.247

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