Let \(\Omega\) be a bounded open subset of \({\mathbb R}^2\), and let \(g\in L^\infty(\Omega)\). The paper proposes an approximation, in the sense of \(\Gamma\)-convergence, of the functional \[ {\mathcal G}(u,C,P)=\#(P)+\int_C(1+\kappa^2)\,d{\mathcal H}^1+\int_{\Omega\backslash(C\cup P)}| \nabla u| ^2\,dx+\int_\Omega| u-g| ^2\,dx, \] where \(C\) is a family of curves, \(P\) is the set of the endpoints of the curves of \(P\), \(\#(P)\) is the number of points in \(P\), \(\kappa\) is the curvature, and \({\mathcal H}^1\) is the one-dimensional Hausdorff measure.
The approximating functionals are of “elliptic type”, so, at least in principle, numerically more tractable. They are of the form \[ \begin{split} {\mathcal G}_\varepsilon(u,s,w)={1\over4\pi b_0}\int_\Omega\left({1\over\varepsilon}+\varepsilon\left(\text{div}{\nabla w\over| \nabla w| }\right)^2\right)\left(\zeta_\varepsilon| \nabla w| ^2+{w^2(1-w)^2\over\zeta_\varepsilon}\right)\,dx+\\ +{1\over2b_0}\int_\Omega w^2\left(1+\left(\text{div}{\nabla s\over| \nabla s| }\right)^2\right)\left(\zeta_\varepsilon| \nabla s| ^2+{s^2(1-s)^2\over\zeta_\varepsilon}\right)\,dx+\int_\Omega s^2| \nabla u| ^2\,dx+\\+\int_\Omega| u-g| ^2dx+{1\over\mu_\varepsilon}\int_\Omega((1-s)^2+(1-w)^2)dx,\end{split} \] for suitable \(\zeta_\varepsilon\) and \(\mu_\varepsilon\) tending to 0, and \(b_0>0\).
Connections with a related conjecture by E. De Giorgi are also discussed.