Lauro Silini’s research while affiliated with ETH Zurich and other places

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Publications (7)


Free energy minimizers with radial densities: classification and quantitative stability
  • Preprint

December 2024

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6 Reads

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Lauro Silini

We study the isoperimetric problem with a potential energy g in Rn\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=0g = 0, the condition ln(f)+g0\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.


Quantitative C 1 -stability of spheres in rank one symmetric spaces of non-compact type

November 2024

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2 Reads

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1 Citation

Advances in Calculus of Variations

We prove that in any rank one symmetric space of non-compact type M ∈ { ℝ ⁢ H n , ℂ ⁢ H m , ℍ ⁢ H m , 𝕆 ⁢ H 2 } {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 {C^{1}} -volume preserving perturbations. We quantify the gain of perimeter in terms of the W 1 , 2 {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.


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The spherical symmetrization
The curl described in Lemma 3.2
The curvature comparison
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Approaching the isoperimetric problem in HCmHCmH_{{\mathbb {C}}}^m via the hyperbolic log-convex density conjecture
  • Article
  • Full-text available

November 2023

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14 Reads

Calculus of Variations and Partial Differential Equations

We prove that geodesic balls centered at some base point are uniquely isoperimetric sets in the real hyperbolic space HRnHRnH_{{\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 RnRn{\mathbb {R}}^n. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are uniquely isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.

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Quantitative C1C^1-stability of spheres in rank one symmetric spaces of non-compact type

April 2023

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11 Reads

We prove that in any rank one symmetric space of non-compact type M{RHn,CHm,HHm,OH2}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 C1C^1-volume preserving perturbations. We quantify the gain of perimeter in terms of the W1,2W^{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.


Approaching the isoperimetric problem in HCmH^m_{\mathbb{C}} via the hyperbolic log-convex density conjecture

July 2022

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5 Reads

We prove that geodesic balls centered at some base point are isoperimetric in the real hyperbolic space HRnH_{\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 Rn\mathbb{R}^n. As an application we prove that in the complex and quaterionic hyperbolic spaces, geodesic balls are isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.


Local existence of smooth solutions for the semigeostrophic equations on curved domains

June 2022

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1 Read

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 R2\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.

Citations (1)


... This question, rooted in the seminal work of Fuglede [30], has a rich history in the study of isoperimetric problems. Rigorous results addressing similar questions have been established in various contexts, including the Euclidean setting [18,19,31,32,34,37], Riemannian manifolds [8,9,13,16,24,62], anisotropic energies [23,27,55,56], and weighted settings [17,20,33]. ...

Reference:

Free energy minimizers with radial densities: classification and quantitative stability
Quantitative C 1 -stability of spheres in rank one symmetric spaces of non-compact type
  • Citing Article
  • November 2024

Advances in Calculus of Variations