\(\Gamma\)-convergence for the irrigation problem. (English) Zbl 1076.49024

Mosconi, Sunra J. N.; Tilli, Paolo

J. Convex Anal. 12, No. 1, 145-158 (2005).

Let \(\Omega\) be a bounded, connected, open set with Lipschitz boundary of \({\mathbb R}^d\) for \(d\geq2\), and let \(\Sigma(\Omega)\) denote the class of all the compact, connected subsets \(\gamma\) of \(\overline\Omega\) of finite one-dimensional Hausdorff measure \({\mathcal H}^1(\gamma)\). In the paper the asymptotic behaviour of a variant of the so-called “irrigation problem” is studied.
Let \(f\in L^1(\Omega)\) be nonnegative. For \(l>0\), let \(F_l\) be the functional defined by \[ F_l(\gamma)=\begin{cases} l^{p\over d-1}\int_\Omega f(x)d_\gamma(x)^pdx &\text{if }\gamma\in\Sigma(\Omega)\text{ and }{\mathcal H}^1(\gamma)=l\\ +\infty &\text{otherwise,}\end{cases} \] where \(d_\gamma\) denotes the distance function to the set \(\gamma\), and \(p>0\) is given.
In order to study the asymptotic behaviour of \(F_l\) as \(l\) goes to \(+\infty\), \(F_l\) has been extended to the space \({\mathcal P}(\overline\Omega)\) of the probability measures supported in \(\overline\Omega\) as \[ F_l(\mu)=\begin{cases} l^{p\over d-1}\int_\Omega f(x)d_\gamma(x)^pdx &\text{if }\mu=l^{-1}{\mathcal H}^1{}_{| \gamma}\text{ for some }\gamma\in\Sigma(\Omega)\text{ such that }{\mathcal H}^1(\gamma)=l\\ +\infty &\text{otherwise.}\end{cases} \] Then, it has been proved that, as \(l\to+\infty\), the \(F_l\)s \(\Gamma\)-converge, with respect to the weak* topology on \({\mathcal P}(\overline\Omega)\), to the functional \(F_\infty\) given by \[ F_\infty(\mu)=\theta_{d,p}\int_\Omega{f(x)\over\varrho(x)^{p\over d-1}}dx, \] where \(\varrho\in L^1(\Omega)\) is the Radon-Nikodým derivative of \(\mu\) with respect to Lebesgue measure, and \(\theta_{d,p}\) is a positive constant, depending only on \(d\) and \(p\), given by an explicit formula.
Since \(F_\infty\) has the unique minimizer in \({\mathcal P}(\overline\Omega)\) given by \(\mu=(f^{(d-1)\over(p+d-1)}/\int_\Omega f^{(d-1)\over(p+d-1)})dx\), it follows that \[ \lim_{l\to+\infty}\min_{\gamma\in\Sigma(\Omega)}F_l(\gamma)=\theta_{d,p}\left(\int_\Omega f(x)^{(d-1)\over(p+d-1)}dx\right)^{p+d-1\over d-1}. \] Moreover, if \(\gamma_n\) is a minimizer of \(F_{l_n}\), and \(l_n\) diverges, then the probability measures \({\mathcal H}^1(\gamma_n)^{-1}\cdot{\mathcal H}^1{}_{| \gamma_n}\) converge in the weak* topology to the probability measure \(\mu=(f^{(d-1)\over(p+d-1)}/\int_\Omega f^{(d-1)\over(p+d-1)})dx\).

Reviewer: Riccardo De Arcangelis (Napoli)

MSC:

49Q20	Variational problems in a geometric measure-theoretic setting
49J45	Methods involving semicontinuity and convergence; relaxation
49Q10	Optimization of shapes other than minimal surfaces

Keywords:

irrigation problem; \(\Gamma\)-convergence; optimal transport problems