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Topological equivalence of some variational problems involving distances. (English) Zbl 1025.49013

Discrete Contin. Dyn. Syst. 7, No. 2, 247-258 (2001); erratum ibid 18, No. 1, 219-220 (2007).
Several kinds of variational problems associated to distance functions on an open subset \(\Omega\) of \({\mathbb R}^N\) are considered in the paper. In particular, given a sequence of distances \(\{d_n\}\) on \(\Omega\), variational problems are considered for the functional \[ L_n(\gamma)=\int_0^1\varphi_{d_n}(\gamma(t),\gamma'(t))dt \] relative to the length induced by the Finsler metric \(\varphi_{d_n}\) associated by differentiation to the distance \(d_n\), where \(\gamma\) is a Lipschitz curve in \(\Omega\), for the functional \[ J_n(\mu)=\int_{\Omega\times\Omega}d_n(x,y)d\mu(x,y) \] coming from mass transfer problems, where \(\mu\) is a positive measure on \(\Omega\times\Omega\), and for the functional \[ F_n(u)=\begin{cases} 0 &\text{ if } \nabla u(x)\in K_n(x) \text{ for }{\mathcal L}^N-\text{a.e. }x\in\Omega \cr +\infty &\text{ otherwise }\end{cases} \] associated by duality to \(J_n\), where \(K_n(x)\) is a convex set associated to \(d_n\) and \(u\) is a Lipschitz function.
The main result of the paper states the equivalence between the \(\Gamma\)-convergence of the functionals \(L_n\), \(J_n\), \(F_n\) and the uniform convergence of \(\{d_n\}\) on compact subsets of \({\mathbb R}^{2N}\). Such equivalences are proved under the assumption that each \(d_n\) belongs to the class of all geodesic distances on \(\Omega\), which are controlled from above and below by fixed multiples of the Euclidean distance.
Examples are also discussed proving that the above result become false out of the above described class of distances.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
49J10 Existence theories for free problems in two or more independent variables
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