Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global 1/(n+1)-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to 1/2 in the case of partitions with the same anisotropy and the same mobility and provide a weak comparison result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

Minimizing movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimizing movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ ≪ ε and slow time scales ε ≪ τ have been investigated in [4]. In this paper, the intermediate (critical) case of finite ratio ε/τ > 0 is studied. It is shown that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. A characterization of the pinning threshold is given. For initial data above the pinning threshold, the equation and velocity describing the homogenized motion are determined.

We prove existence of minimizing movements for thedislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscositysolutions of the corresponding level-set equation. We also prove theconsistency of this approach, by showing that any minimizing movementcoincides with the smooth evolution as long as the latter exists. Inrelation with this, we finally prove short time existence and uniqueness of a smoothfront evolving according to our law, provided the initial shape issmooth enough.

This paper addresses the Cauchy problem for thegradient flow equation in a Hilbert space  $\mathcal{H}$ \[\begin{cases}u'(t)+ \partial_{\ell}\phi(u(t))\ni f(t)&\text{{\it a.e.}\ in }(0,T),u(0)=u_0, \end{cases}\] where $\phi: \mathcal{H} \to (-\infty,+\infty]$ is a proper,lower semicontinuous functional which is not supposed to be a(smooth perturbation of a) convex functional and $\partial_{\ell}\phi$ is(a suitable limiting version of) its subdifferential.We will present some new existence results for the solutions of theequation by exploiting a variational approximationtechnique, featuring some ideas from the theory of Minimizing Movementsand of Young measures.
Our analysisis also motivated by some models describing phase transitionsphenomena, leading tosystems of evolutionary PDEs which have a common underlying gradient flow structure:in particular, we will focus onquasistationary models, which exhibithighly non convex Lyapunov functionals.