arXiv subjects
Publications and source records attributed to Lauro Silini.
We prove a Heintze--Karcher type inequality for a large class of log-convex weights in Euclidean and Hyperbolic spaces. As a consequence we obtain an Alexandrov theorem for $λ$-self expanders.
We study the isoperimetric problem with a potential energy $g$ in $\mathbb{R}^n$ weighted by a radial density $f$ and analyze the geometric properties of minimizers. Notably, we construct two counterexamples demonstrating that, in contrast to the classical isoperimetric case $g = 0$, the condition $\ln(f)'' + g' \geq 0$ does not generally guarantee the global optimality of centered spheres. However, we demonstrate that centered spheres are globally optimal when both $f$ and $g$ are monotone. Additionally, we strengthen this result by deriving a sharp quantitative stability inequality.
We prove that in any rank one symmetric space of non-compact type $M\in\{\mathbb{R} H^n,\mathbb{C} H^m,\mathbb{H} H^m,\mathbb{O} H^2\}$, geodesic spheres are uniformly quantitatively stable with respect to small $C^1$-volume preserving perturbations. We quantify the gain of perimeter in terms of the $W^{1,2}$-norm of the perturbation, taking advantage of the explicit spectral gap of the Laplacian on geodesic spheres in $M$. As a consequence, we give a quantitative proof that for small volumes, geodesic spheres are isoperimetric regions among all sets of finite perimeter.
We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space $H_{\mathbb R}^n$ endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on $\mathbb R^n$. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.
We prove local-in-time existence of smooth solutions to the semigeostrophic equations in the general setting of smooth, bounded and simply connected domains of $\mathbb{R}^2$ endowed with an arbitrary conformally flat metric and non-vanishing Coriolis term. We present a construction taking place in Eulerian coordinates, avoiding the classical reformulation in dual variables, used in the flat case with constant Coriolis force, but lacking in this general framework.