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Virginia Agostiniani

Publications and source records attributed to Virginia Agostiniani.

At least 19 recordsLinked to original sources
arXiv2026

Estimates for the p-Green function near the pole

We study the asymptotic expansion of p-Green functions and their derivatives near the pole. In the Euclidean setting, we strengthen a theorem from [arXiv:2203.01206] by establishing improved integrability properties and estimates for the first and second derivatives. In the more general Riemannian setting, we derive a refined asymptotic expansion of the p-Green function near its pole, thereby improving the corresponding result proved in [arXiv:2512.14591].

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arXiv2026

Riemannian Penrose Inequality for Manifolds with Corners via Non-Linear Potential Theory

We present a new proof of the Positive Mass Theorem and the Riemannian Penrose Inequality for three-dimensional asymptotically flat Riemannian manifolds whose metrics fail to be $C^1$ across a hypersurface $\Sigma$, first proven by Miao and McCormick-Miao, respectively. Unlike these approaches, ours recovers these results directly, without relying on their original formulations for smooth metrics. The proofs are based on a unified argument which applies to both theorems. We achieve this by establishing an approximate monotonicity for the quantity introduced by Agostiniani-Mantegazza-Mazzieri-Oronzio, employing the approximation scheme of Miao, for metrics with $C^{2,\alpha}$ regularity up to $\Sigma$.

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arXiv2026

Singularly Perturbed Gradient Flows and Evolution of Critical Points in Infinite Dimensions

We consider singularly perturbed gradient flows in Hilbert spaces, driven by a time-dependent, nonconvex, and nonsmooth energy, and address the convergence of their solutions to curves of critical points of the driving energy functional. The degenerating nature of the estimates along the gradient-flow curves calls for novel compactness arguments, which we carefully develop by combining tools from the variational approach to Hilbert and metric gradient flows \cite{RossiSavare06,AGS08}, with fine requirements on the set of critical points of the energy. This leads us to prove that subsequential limits of singularly perturbed gradient flows are Dissipative Viscosity solutions of the limiting problem, i.e., curves of critical points satisfying a suitable balance between the energy and a defect measure, encoding dissipation. This energy-dissipation balance encompasses information on the dynamics of the process at jump times, recording, in particular, the re-emergence of viscous behavior. Under a suitable rectifiability condition on the critical set, we show that \DSolutions\ improve to Balanced Viscosity solutions, which have the key property that the dissipation measure is purely atomic. In the second part of the paper we show that, for smooth energies whose second differential is a Fredholm operator, the condition that the kernel of the Hessian has dimension at most one at every critical point already implies our measure-theoretic assumptions. We further relate them to the transversality conditions from bifurcation theory and show that they have a generic character with respect to a wide family of linear perturbations of the energy. The results are new also in finite dimension. Applications to various PDE models are presented.

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arXiv2026

Mass-type invariants in the presence of a cosmological constant

In this paper, we introduce a new family of mass-type invariants for time-symmetric initial data in space-times satisfying the Dominant Energy Condition. For positive cosmological constant, these invariants, unlike the total Hawking mass, turn out to be genuinely effective in providing new characterizations of the de Sitter solution. From a theoretical standpoint, this opens a new perspective on how one might refine the rigidity statement originally proposed by Min-Oo in his well known conjecture, later refuted by the counterexamples of Brendle, Marques, and Neves. Via a formal limiting procedure, we also define another invariant, the 1-harmonic Mass, for which we independently prove a positive mass theorem and a Penrose-type inequality, thereby extending tools for probing space-time geometries in the presence of a positive cosmological constant.

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arXiv2025

Riemannian Penrose inequality via Nonlinear Potential Theory

We provide a new proof of the Riemannian Penrose inequality for time-symmetric asymptotically flat initial data with a single black-hole horizon. The proof proceeds through a newly established monotonicity formula holding along the level sets of the $p$-capacitary potential of the horizon boundary, in any asymptotically flat $3$-manifold with nonnegative scalar curvature.

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arXiv2021

On the Serrin problem for ring-shaped domains

In this paper, we deal with the long standing open problem of characterising rotationally symmetric solutions to $Δu = -2$, when Dirichlet boundary conditions are imposed on a ring-shaped planar domain. From a physical perspective, the solution represents the velocity of a homogeneous incompressible fluid, flowing in steady parallel streamlines through a hollow cylindrical pipe and obeying a no-slip condition. In contrast with Serrin's classical result, we show that the simplest possible set of overdetermining conditions, namely the prescription of locally constant Neumann boundary data, is not sufficient to obtain a complete characterisation of the solutions. A further requirement on the number of maximum points arises in our analysis as a necessary and sufficient condition for the rotational symmetry. In fluid-dynamical terms, our results imply that if the wall shear stress is constant on some connected component of the pipe's wall, then the velocity of the fluid must attain its maximal value only at finitely many streamlines, unless the hollow pipe itself consists of a couple of concentric cylindrical round tubes. A major difficulty in the analysis of this problem comes from the lack of monotonicity of the model solutions, which makes the moving plane method ineffective. To remedy this issue, we introduce some new arguments in the spirit of comparison geometry, that we believe of independent interest.

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arXiv2021

Minkowski Inequalities via Nonlinear Potential Theory

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $Ω\subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the $p$-capacitary potentials associated with $Ω$, for every $p$ sufficiently close to $1$. Besides constituting a neat improvement of those introduced in [Fog_Maz_Pin] to treat the case of convex domains, these formulas testify the existence of a link between the monotonicity formulas derived by Colding and Minicozzi for the level set flow of Green's functions and the monotonicity formulas employed by Huisken, Ilmanen and several other authors in studying the geometric implications of the Inverse Mean Curvature Flow. In dimension $n\geq 8$, our conclusions are stronger than the ones obtained so far through the latter mentioned technique.

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arXiv2020

A geometric capacitary inequality for sub-static manifolds with harmonic potentials

In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family $\{F_β\}$ of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold $β=\frac{n-2}{n-1}$ and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.

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arXiv2019

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

In this paper we consider complete noncompact Riemannian manifolds $(M, g)$ with nonnegative Ricci curvature and Euclidean volume growth, of dimension $n \geq 3$. We prove a sharp Willmore-type inequality for closed hypersurfaces $\partial Ω$ in $M$, with equality holding true if and only if $(M{\setminus}Ω, g)$ is isometric to a truncated cone over $\partialΩ$. An optimal version of Huisken's Isoperimetric Inequality for $3$-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue's non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

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arXiv2018

Monotonicity formulas in potential theory

Using the electrostatic potential $u$ due to a uniformly charged body $Ω\subset\mathbb R^n$, $n\geq 3$, we introduce a family of monotone quantities associated with the level set flow of $u$. The derived monotonicity formulas are exploited to deduce a new quantitative version of the classical Willmore inequality.

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arXiv2018

Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets

We rigorously derive a Kirchhoff plate theory, via $Γ$-convergence, from a three-di\-men\-sio\-nal model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables $x'$. At the same time, the spontaneous strain is $h$-close to the identity, where $h$ is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into $\mathbb{R}^3$, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from a $x'$-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.

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arXiv2017

Rigorous derivation of active plate models for thin sheets of nematic elastomers

In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced from a three-dimensional description of the system using rigorous dimension-reduction techniques, based on the theory of Gamma-convergence. The two-dimensional models are nonlinear plate theories in which deviations from a characteristic target curvature tensor cost elastic energy. Moreover, the stored energy functional cannot be minimised to zero, thus revealing the presence of residual stresses, as observed in numerical simulations. The following three nematic textures are considered: splay-bend and twisted orientation of the nematic director, and uniform director perpendicular to the mid-plane of the film, with variable degree of nematic order along the thickness. These three textures realise three very different structural models: one with only one stable spontaneously bent configuration, a bistable one with two oppositely curved configurations of minimal energy, and a shell with zero stiffness to twisting.

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arXiv2017

Dimension reduction via Gamma-convergence for soft active materials

We present a rigorous derivation of dimensionally reduced theories for thin sheets of nematic elastomers, in the finite bending regime. Focusing on the case of twist nematic texture, we obtain 2D and 1D models for wide and narrow ribbons exhibiting spontaneous flexure and torsion. We also discuss some variants to the case of twist nematic texture, which lead to 2D models with different target curvature tensors. In particular, we analyse cases where the nematic texture leads to zero or positive Gaussian target curvature, and the case of bilayers.

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arXiv2016

Singular vanishing-viscosity limits of gradient flows: the finite-dimensional case

In this note we study the singular vanishing-viscosity limit of a gradient flow set in a finite-dimensional Hilbert space and driven by a smooth, but possibly non convex, time-dependent energy functional. We resort to ideas and techniques from the variational approach to gradient flows and rate-independent evolution to show that, under suitable assumptions, the solutions to the singularly perturbed problem converge to a curve of stationary points of the energy, whose behavior at jump points is characterized in terms of the notion of Dissipative Viscosity solution. We also provide sufficient conditions under which Dissipative Viscosity solutions enjoy better properties, which turn them into Balanced Viscosity solutions. Finally, we discuss the generic character of our assumptions.

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arXiv2016

Shape programming for narrow ribbons of nematic elastomers

Using the theory of $Γ$-convergence, we derive from three-dimensional elasticity new one-dimensional models for non-Euclidean elastic ribbons, i.e. ribbons exhibiting spontaneous curvature and twist. We apply the models to shape-selection problems for thin films of nematic elastomers with twist and splay-bend texture of the nematic director. For the former, we discuss the possibility of helicoid-like shapes as an alternative to spiral ribbons.

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arXiv2015

On the geometry of the level sets of bounded static potentials

In this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity. In the case where the static potential is bounded, we introduce new quantities which are proven to be monotone along the level set flow of the potential function. We then show how to use these properties to detect the rotational symmetry of the static solutions, deriving a number of sharp inequalities. As a consequence of our analysis, a simple proof of the classical $3$-dimensional Black Hole Uniqueness Theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions.

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arXiv2015

Riemannian aspects of potential theory

In this paper we provide a new method for establishing the rotational symmetry of the solutions to a couple of very classical overdetermined problems arising in potential theory, in both the exterior and the interior punctured domain. Thanks to a conformal reformulation of the problems, we obtain Riemannian manifolds with zero Weyl tensor satisfying a quasi-Einstein type equation. Exploiting these geometric properties, we conclude via a splitting argument that the manifolds obtained are half cylinders. In turn, the rotational symmetry of the potential is implied. To the authors' knowledge, some of the overdetermining conditions considered here are new.

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