A Gradient Flow of the One-Dimensional Alt-Caffarelli Functional

We construct a candidate of the gradient flow for a variational functional with a singular term in the one-dimensional case. Applied is the scheme using the theory of Γ-convergence for variational functionals, where both the Dirichlet integral and the singular term are approximated at the same time. An approximated solution is constructed on the space of piecewise constant functions, and the desired gradient flow is built as the limit of it. We establish some properties of the gradient flow as well as obtain PDE including a term of integration by Radon measure.