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Filomena Feo

Publications and source records attributed to Filomena Feo.

15 recordsLinked to original sources
arXiv2025

Some obstacle problems for partially hinged plates and related optimization issues

We study optimization problems for partially hinged rectangular plates, modeling bridge roadways, in the presence of real and artificial obstacles. Real obstacles represent structural constraints to avoid, while artificial ones are introduced to enhance stability. For the former, aiming to prevent collisions, we set up a worst-case optimization problem in which we minimize the amplitude of oscillations with respect to the density distribution; for the latter, aiming to improve the torsional stability, we minimize, with respect to the obstacles, the maximum of a gap function quantifying the displacement between the long edges of the plate. For both problems, existence results are provided, along with a discussion about qualitative properties of optimal density distributions and obstacles.

math.OC↗
arXiv2024

Asymptotic behaviour of solutions and free boundaries of the anisotropic slow diffusion equation

In this paper we explore the theory of the anisotropic porous medium equation in the slow diffusion range. After revising the basic theory, we prove the existence of self-similar fundamental solutions (SSFS) of the equation posed in the whole Euclidean space. Each of such solutions is uniquely determined by its mass. This solution has compact support w.r.t. the space variables. We also obtain the sharp asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Special attention is paid to the convergence of supports and free boundaries in relative size, i.e., measured in the appropriate anisotropic way. The fast diffusion case has been studied in a previous paper by us, there no free boundaries appear.

math.AP↗
arXiv2022

Another approach to the equality case of the sharp logarithmic Sobolev inequality

In this note, we characterize the equality case of the sharp $L^2$-Euclidean logarithmic Sobolev inequality with monomial weights, exploiting the idea by Bobkov and Ledoux \cite{Bob}. Our approach is new even in the unweighted case. Also, we show that the same strategy yields the equality case of the sharp $L^p$-Euclidean logarithmic Sobolev inequality for $1 < p < \infty$ with arbitrary norm on ${\mathbb R}^n$, if the inequality is unweighted.

math.AP↗
arXiv2021

Anisotropic p-Laplacian Evolution of Fast Diffusion type

We study an anisotropic, possibly non-homogeneous version of the evolution $p$-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive $L^1$ to $L^\infty$ estimates. We prove the existence of a self-similar fundamental solution of this equation in the appropriate exponent range, and uniqueness in a smaller range. We also obtain the asymptotic behaviour of finite mass solutions in terms of the self-similar solution. Positivity, decay rates as well as other properties of the solutions are derived. The combination of self-similarity and anisotropy is not common in the related literature. It is however essential in our analysis and creates mathematical difficulties that are solved for fast diffusions.

math.AP↗
arXiv2020

Anisotropic Fast Diffusion Equations

We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite-mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The combination of self-similarity and anisotropy is essential in our analysis and creates serious mathematical difficulties that are addressed by means of novel methods.

math.AP↗
arXiv2019

Sobolev Anisotropic inequalities with monomial weights

We derive some anisotropic Sobolev inequalities in $\mathbb{R}^{n}$ with a monomial weight in the general setting of rearrangement invariant spaces. Our starting point is to obtain an integral oscillation inequality in multiplicative form.

math.FA↗
arXiv2019

Sharp Logarithmic Sobolev and related inequalities with monomial weights

We derive a sharp Logarithmic Sobolev inequality with monomial weights starting from a sharp Sobolev inequality with monomial weights. Several related inequalities such as Shannon type and Heisenberg's uncertain type are also derived. A characterization of the equality case for the Logarithmic Sobolev inequality is given when the exponents of the monomial weights are all zero or integers. Such a proof is new even in the unweighted case.

math.AP↗
arXiv2016

Comparison results for nonlinear anisotropic parabolic problems

Comparison results for solutions to the Dirichlet problems for a class of nonlinear, anisotropic parabolic equations are established. These results are obtained through a semi-discretization method in time after providing estimates for solutions to anisotropic elliptic problems with zero-order terms.

math.AP↗