Topological equivalence of some variational problems involving distances

Giuseppe Buttazzo; Luigi De Pascale; Ilaria Fragalà

doi:10.3934/dcds.2001.7.247

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2001, Volume 7, Issue 2: 247-258. Doi: 10.3934/dcds.2001.7.247

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Topological equivalence of some variational problems involving distances

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

Revised: July 31, 2000

Published: March 2000

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Abstract

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.

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Mathematics Subject Classification: 49J45, 53C60.

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