Stationary sets of the mean curvature flow with a forcing term

We consider the flat flow solution to the mean curvature equation with forcing in ℝ n {\mathbb{R}^{n}} . Our main result states that tangential balls in ℝ n {\mathbb{R}^{n}} under a flat flow with a bounded forcing term will experience fattening, which generalizes the result in [N. Fusco, V. Julin and M. Morini, Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane, preprint 2020, https://arxiv.org/abs/2004.07734] from the planar case to higher dimensions. Then, as in the planar case, we characterize stationary sets in ℝ n {\mathbb{R}^{n}} for a constant forcing term as finite unions of equisize balls with mutually positive distance.

Prescribed mean curvature equation on torus

Prescribed mean curvature problems on the torus have been considered in one dimension. In this paper, we prove the existence of a graph on the n-dimensional torus 𝕋 n {\mathbb{T}^{n}} , the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.

New Bernstein Type Results in Weighted Warped Products

In this paper, we obtain new parametric uniqueness results for complete constant weighted mean curvature hypersurfaces under suitable geometric assumptions in weighted warped products. Furthermore, we also prove very general Bernstein type results for the constant mean curvature equation for entire graphs in these ambient spaces.

Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{Ω},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{Ω},\end{array}\right. with \text{Ω} an open ball in {{\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.