American Institute of Mathematical Sciences

Advanced
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

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
[1]

Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427
[2]

Feng Luo. Geodesic length functions and Teichmuller spaces. Electronic Research Announcements, 1996, 2: 34-41.

[3]

Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017
[4]

Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679
[5]

Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 411-426. doi: 10.3934/dcds.2015.35.411
[6]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071
[7]

J. M. Mazón, Julio D. Rossi, J. Toledo. Optimal matching problems with costs given by Finsler distances. Communications on Pure & Applied Analysis, 2015, 14 (1) : 229-244. doi: 10.3934/cpaa.2015.14.229
[8]

Guillaume Bal, Olivier Pinaud, Lenya Ryzhik. On the stability of some imaging functionals. Inverse Problems & Imaging, 2016, 10 (3) : 585-616. doi: 10.3934/ipi.2016013
[9]

P. Di Gironimo, L. D’Onofrio. On the regularity of minimizers to degenerate functionals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1311-1318. doi: 10.3934/cpaa.2010.9.1311
[10]

Jean-François Babadjian, Francesca Prinari, Elvira Zappale. Dimensional reduction for supremal functionals. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1503-1535. doi: 10.3934/dcds.2012.32.1503
[11]

Dmitry Dolgopyat, Dmitry Jakobson. On small gaps in the length spectrum. Journal of Modern Dynamics, 2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339
[12]

Victor Berdichevsky. Distribution of minimum values of stochastic functionals. Networks & Heterogeneous Media, 2008, 3 (3) : 437-460. doi: 10.3934/nhm.2008.3.437
[13]

Guillaume Bal, Chenxi Guo, Francçois Monard. Linearized internal functionals for anisotropic conductivities. Inverse Problems & Imaging, 2014, 8 (1) : 1-22. doi: 10.3934/ipi.2014.8.1
[14]

Menita Carozza, Gioconda Moscariello, Antonia Passarelli. Higher integrability for minimizers of anisotropic functionals. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 43-55. doi: 10.3934/dcdsb.2009.11.43
[15]

Patrick Cummings, C. Eugene Wayne. Modified energy functionals and the NLS approximation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1295-1321. doi: 10.3934/dcds.2017054
[16]

Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51
[17]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109
[18]

Osama Khalil. Geodesic planes in geometrically finite manifolds. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 881-903. doi: 10.3934/dcds.2019037
[19]

Yong Fang. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenst$\ddot{a}$dt distances. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3471-3483. doi: 10.3934/dcds.2014.34.3471
[20]

Jonathan Zinsl. The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 919-933. doi: 10.3934/dcdss.2017047

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences