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A regularity for a parabolic p-Laplacian system $$(p>2)$$ is studied by the use of the discrete Morse flow method which is known as one of the ways to approximate a solution to parabolic partial differential equations. Our approximate solution is constructed from the sequence of minimizers of variational functionals whose Euler–Lagrange equations are the time discretized p-Laplacian system. The aim of this paper is to establish that the regularity estimates for the approximate solution hold uniformly on two approximation parameters and show strong convergence of the approximate solution.

By means of a minimizing scheme, we constructively define approximate solutions to parabolic systems. On the basis of Campanato theory, these approximate solutions are shown to be locally Holder equi-continuous with respect to an approximation parameter. As an application, we obtain a weak solution to an initial-boundary value problem with the same Holder continuity.

The present paper shows Holder continuity of discrete Morse flows to a regularized Alt–Caffarelli variational functional generating free boundaries; the continuity is uniform with respect to the discrete Morse flows and the regularizations. The uniformity enables to construct Morse flows to the Alt–Caffarelli functional, which shall be dealt with in another paper.

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.

The free boundary problem of the variational type studied by Alt and Caffarelli is investigated here. Especially, treated is the quadratic functional having the coefficient of diagonal form and the characteristic function term. The main result is to construct a minimizer with the Lipschitz continuity, which will play an essential role for proving the regularity of the free boundary. This is established by treating the regularized functional which approximates the original functional in the sense of Γ-convergence. The most important part of this paper is to show the uniform Lipschitz continuity of the minimizers of the regularized functionals. Taking advantage of the minimality, not resorting to the Euler–Lagrange equation, the minimizers turn out to belong to DeGiorgi class uniformly, the uniform Holder estimate is obtained, and, as a result, the Lipschitz estimate is attained with the aid of the uniform Harnack inequality.

We treat a variational problem with a singular term by the method of approximation via Γ-convergence, which is a constructive way. It is known that the minimizers may have a free boundary in the interior of the domain. The main aim of this paper is to establish a criterion on the location of the free boundary in terms of the approximation. As a corollary, we obtain a non-degeneracy property which means that the convergence of the approximated free boundary is sharp in some sense, and show the corresponding examples in a one-dimensional linear case. In order to derive the non-degeneracy property we need a device to the choice of approximation to the singular term.