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

Topological equivalence of some variational problems involving distances

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.
Citation: Giuseppe Buttazzo, Luigi De Pascale, Ilaria Fragalà. Topological equivalence of some variational problems involving distances. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427

[2]

Lorenza D'Elia. $ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices. Networks and Heterogeneous Media, 2022, 17 (1) : 15-45. doi: 10.3934/nhm.2021022

[3]

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

[4]

Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355

[5]

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

[6]

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

[7]

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

[8]

Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2101-2116. doi: 10.3934/cpaa.2021059

[9]

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

[10]

Harun Karsli, Purshottam Narain Agrawal. Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022002

[11]

Nhan-Phu Chung. Gromov-Hausdorff distances for dynamical systems. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6179-6200. doi: 10.3934/dcds.2020275

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Emmanuel Schenck. Exponential gaps in the length spectrum. Journal of Modern Dynamics, 2020, 16: 207-223. doi: 10.3934/jmd.2020007

[18]

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

[19]

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

[20]

Alice Le Brigant. Computing distances and geodesics between manifold-valued curves in the SRV framework. Journal of Geometric Mechanics, 2017, 9 (2) : 131-156. doi: 10.3934/jgm.2017005

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (171)
  • HTML views (0)
  • Cited by (3)

[Back to Top]