We prove a \(C^{1,1}\)-regularity of minimizers of the functional \[ \int_I \sqrt{1+\vert Du\vert ^2} + \int_I \vert u-g\vert\,ds,\quad u\in BV(I), \] provided \(I\subset\mathbb{R}\) is a bounded open interval and \(\vert\vert g\vert\vert_\infty\) is sufficiently small, thus partially establishing a De Giorgi conjecture in dimension one and codimension one. We also extend our result to a suitable anisotropic setting.

In this paper we prove that in $\mathbb R^3$ the minimizing movement solutions for mean curvature motion of droplets, obtained in [Bellettini, Kholmatov: J. Math. Pures Appl. (2018)|https://www.sciencedirect.com/science/article/pii/S0021782418300825 ], starting from a regular droplets sitting on the horizontal plane with a regular relative adhesion coefficient, coincide with the smooth mean curvature flow of droplets with a prescribed contact-angle.

We study the crystalline curvature flow of planar networks with a single hexagonal anisotropy. After proving the local existence of a classical solution for a rather large class of initial conditions, we classify the homothetically shrinking solutions having one bounded component. We also provide an example of network shrinking to a segment with multiplicity two.

In this paper we prove that the Winterbottom shape [[Winterbottom: Acta Metallurgica (1967)]] is a volume-constraint minimizer of the corresponding anisotropic capillary functional.

We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in [[Bellettini, Kholmatov: J. Math. Pures Appl (2018)]] we establish the existence of smooth flow, starting from a regular droplet and satisfying the prescribed anisotropic Young's law, and also the existence of a $1/2$-Hölder continuous in time minimizing movement solution starting from a set of finite perimeter. Furthermore, we investigate various properties of minimizing movements, including comparison principles, uniform boundedness and the consistency with the smooth flow.

Motivated by a conjecture of De Giorgi, we consider the Almgren-Taylor-Wang scheme for mean curvature flow, where the volume penalization is replaced by a term of the form \[ \int_{E\Delta F} f\Big(\frac{ {\rm d}_F }{\tau}\Big)~dx \] for $f$ ranging in a large class of strictly increasing continuous functions. In particular, our analysis covers the case \[ f(r) = r^\alpha, \qquad r \geq 0, \quad \alpha>0, \] considered by De Giorgi. We show that the generalized minimizing movement scheme converges to the geometric evolution equation \[ f(v) = - \kappa\quad \text{on $\partial E(t)$,} \] where $\{E(t)\}$ are evolving subsets of $\mathbb{R}^n,$ $v$ is the normal velocity of $\partial E(t),$ and $\kappa$ is the mean curvature of $\partial E(t)$. We extend our analysis to the anisotropic setting, and in the presence of a driving force. We also show that minimizing movements coincide with the smooth classical solution as long as the latter exists. Finally, we prove that in the absence of forcing, mean convexity and convexity are preserved by the weak flow.