Hölder continuity of discrete Morse flows to the Alt-Caffarelli variational problem generating free boundaries

The present paper shows Hölder 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.