Short Communications

In this talk I will discuss some results obtained in collaboration with G. Staffilani under the GHAIA project. These results include the validity of some a priori estimates for Schrödinger equations with variable coefficients, and the local well-posedness of the corresponding nonlinear initial value problem.

We review recent results on critical points of the sub-Finsler area in the first Heisenberg group, including Pansu-Wulff shapes, regularity of surfaces with prescribed mean curvature, minimizing cones and Bernstein type problems. These results have been obtained in collaboration with J. Pozuelo, M. Ritoré.

In the Heisenberg group $\mathbb{H}^n$ we consider a notion of perimeter associated to a convex body $K\subset \mathbb{R}^{2n}$ of class $C^2_+$ containing $0$ in its interior. By means of the first variation formula, we define a natural notion of sub-Finsler mean curvature of a $t$-graph and study the sub-Finsler prescribed mean curvature equation on a bounded domain $\Omega$ in the hyperplane $\{t=0\}$ that results when imposing that the vertical graph of a function over $\Omega$ is a hypersurface with prescribed mean curvature $H$. When $H$ is constant and strictly smaller than the anisotropic mean curvature of $\partial \Omega$, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of an approximation scheme. These results were obtained in a joint work with Giovannardi, Pinamonti and Verzellesi.