A gradient flow of the one-dimensional Alt-Caffarelli functional
\textit{H.~W.~Alt} and \textit{L.~A.~Caffarelli} [J. Reine Angew. Math. 325, 105--144 (1981; Zbl 0449.35105)] introduced the functional \(AC\) defined as \(AC(u)= \int_{\mathbb R}(|\nabla u|^2+\xi_{\{u>0\}}) \,d{\mathcal L}^1\), where \(\xi_{\{u>0\}}\) is the characteristic function of the set. The gradient flow for the functional \(AC\) is given as a solution of the initial value problem: \(u'(t) = -[\nabla AC](u(t))\), \(u(0)=u_0\), where \(u_0\) is an initial data. In this paper, the author constructs a candidate of the gradient flow for a variational functional with a singular term in the one-dimensional case. The scheme, using the theory of \(\Gamma\)-convergence for variational functionals, where both the Dirichlet integral and the singular term are approximated at the same time, is applied. The author constructs a sequence of approximated functionals \(AC_{\varepsilon}\) defined on the set \(PC^2_{\varepsilon}(\mathbb R)\) of \(\varepsilon\)-meshed piecewise constant \(L^2(\mathbb R)\) functions which converges as \(\varepsilon\to 0\) to the original functional \(AC\) in the sense of \(\Gamma\)-convergence. Some properties of the gradient flow are established and the PDE including a term of integration by Radon measure is obtained.
scientific article; zbMATH DE number 5292347
scientific article