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9 results on '"Volzone, B."'

2. Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics

Author

Carrillo, J. A., Hittmeir, S., Volzone, B., and Yao, Y.

Subjects
Mathematics - Analysis of PDEs
Abstract

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as $t\to\infty$., Comment: Fix a small gap in the proof of Proposition 2.15 leading to Case 2 of this proof

Published
2016

3. Fractional semilinear Neumann problems arising from a fractional Keller--Segel model

Author

Stinga, P. R. and Volzone, B.

Subjects
Mathematics - Analysis of PDEs, Mathematics - Classical Analysis and ODEs
Abstract

We consider the following fractional semilinear Neumann problem on a smooth bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, $$\begin{cases} (-\varepsilon\Delta)^{1/2}u+u=u^{p},&\hbox{in}~\Omega,\\ \partial_\nu u=0,&\hbox{on}~\partial\Omega,\\ u>0,&\hbox{in}~\Omega, \end{cases}$$ where $\varepsilon>0$ and $1

Published
2014

5. Symmetrization for fractional nonlinear elliptic problems

Author

Ferone, V, Volzone, B, Ferone, Vincenzo, and Volzone, Bruno

Subjects
Symmetrization, fractional Laplacian, nonlocal elliptic equations, Mathematics - Analysis of PDEs, FOS: Mathematics, 35B45, 35R11, 35J25, Analysis of PDEs (math.AP)
Abstract

In this note we prove a new symmetrization result, in the form of mass concentration comparison, for solutions of nonlocal nonlinear Dirichlet problems involving fractional p Laplacians. Some regularity estimates of solutions will be established as a direct application of the main result., Comment: Accepted for publication on Discrete and Continuous Dynamical Systems

Published
2022

6. Bourgain-Brezis-Mironescu formula for magnetic operators

Author

Squassina, Marco, Volzone, B., Squassina, Marco (ORCID:0000-0003-0858-4648), Squassina, Marco, Volzone, B., and Squassina, Marco (ORCID:0000-0003-0858-4648)

Abstract

We prove a Bourgain–Brézis–Mironescu-type formula for a class of nonlocal magnetic spaces, which builds a bridge between a fractional magnetic operator recently introduced and the classical theory.

Published
2016

7. Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction

Author

Carrillo, Jos�� Antonio, Castorina, Daniele, Volzone, Bruno, Carrillo, Ja, Castorina, D, and Volzone, B

Subjects
logarithmic interaction, Mathematics - Analysis of PDEs, FOS: Mathematics, ground states, free energy, logarithmic interaction, uniqueness, uniqueness, ground states, free energy, Analysis of PDEs (math.AP)
Abstract

Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyse the stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy. Furthermore, we prove that this global minimizer is a radially decreasing compactly supported continuous density function which is smooth inside its support, and it is characterized as the unique compactly supported stationary state of the evolution model. This unique profile is the clear candidate to describe the long time asymptotics of the diffusion dominated classical Keller-Segel model for general initial data., 30 pages, 2 figures

Published
2015

8. Comparison and regularity results for the fractional Laplacian via symmetrization methods

Author

Bruno Volzone, Giuseppina di Blasio, DI BLASIO, Giuseppina, and Volzone, B.

Subjects
Fractional Laplacian, Symmetrization, Comparison results, 35R11, 35B45, 15A15, Applied Mathematics, Mathematics - Analysis of PDEs, FOS: Mathematics, Applied mathematics, A priori and a posteriori, Boundary value problem, Laplace operator, Analysis, Analysis of PDEs (math.AP), Mathematics
Abstract

In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well known ones for the classical Laplacian., Comment: 23 pages, 1 figure

Published
2012

9. Ground states in the diffusion-dominated regime.

Author

Carrillo JA, Hoffmann F, Mainini E, and Volzone B

Abstract

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and C ∞ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.

Published
2018
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