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Daniele Bartolucci

Publications and source records attributed to Daniele Bartolucci.

36 records · Page 2Linked to original sources
arXiv2021

On the uniqueness and monotonicity of solutions of free boundary problems

For any $Ω\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on $Ω$ in a neat interval depending only by the best constant of the Sobolev embedding $H^{1}_0(Ω)\hookrightarrow L^{2p}(Ω)$, $p\in [1,\frac{N}{N-2})$ and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for $p>1$, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for $p=1$. The threshold, which is sharp for $p=1$, yields a new condition which guarantees that there is no free boundary inside $Ω$. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes.

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arXiv2020

On the global bifurcation diagram of the Gel'fand problem

For domains of first kind [7,13] we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gel'fand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of non-minimal solutions which is not just concerned with radial solutions [28] and/or with symmetric domains [23]. Toward our goal we parametrize the branch not by the $L^{\infty}(Ω)$-norm of the solutions but by the energy of the associated mean field problem. The proof relies on a carefully modified spectral analysis of mean field type equations.

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arXiv2020

Mean field equations and domains of first kind

In this paper we are interested in understanding the structure of domains of first and second kind, a concept motivated by problems in statistical mechanics. We prove some openness property for domains of first kind with respect to a suitable topology, as well as some sufficient condition for a simply connected domain to be of first kind in terms of the Fourier coefficients of the Riemann map. Finally, we show that the set of simply connected domains of first kind is contractible.

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arXiv2019

Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on a flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions by using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus.

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arXiv2018

Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains

The aim of this paper is to complete the program initiated in [50], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov-Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.

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arXiv2018

Local uniqueness of $m$-bubbling sequences for the Gel'fand equation

We consider the Gel'fand problem, $$ \begin{cases} Δw_{\varepsilon}+\varepsilon^2 h e^{w_{\varepsilon}}=0\quad&\mbox{in}\quadΩ, w_{\varepsilon}=0\quad&\mbox{on}\quad\partialΩ, \end{cases} $$ where $h$ is a nonnegative function in ${Ω\subset\mathbb{R}^2}$. Under suitable assumptions on $h$ and $Ω$, we prove the local uniqueness of $m-$bubbling solutions for any $\varepsilon>0$ small enough.

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arXiv2018

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

We derive a singular version of the Sphere Covering Inequality which was recently introduced in [42], suitable for treating singular Liouville-type problems with superharmonic weights. As an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains, as well as new self-contained proofs of previously known results, such as the uniqueness of spherical convex polytopes first established in [56]. Furthermore, we derive new symmetry results for the spherical Onsager vortex equation.

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arXiv2018

Non degeneracy, Mean Field Equations and the Onsager theory of 2D turbulence

The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require the analysis of a subtle open problem about bubbling solutions of the mean field equation. Motivated by this application we prove that, under suitable non degeneracy assumptions on the associated $m$-vortex Hamiltonian, the $m$-point bubbling solutions of the mean field equation are non degenerate as well. Then we deduce that the Onsager mean field equilibrium entropy is smooth and strictly convex in the high energy regime on domains of second kind.

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arXiv2018

Uniqueness of bubbling solutions of mean field equations

We prove uniqueness of blow up solutions of the mean field equation as $ρ_n \rightarrow 8πm$, $m\in\mathbb{N}$. If $u_{n,1}$ and $u_{n,2}$ are two sequences of bubbling solutions with the same $ρ_n$ and the same (non degenerate) blow up set, then $u_{n,1}=u_{n,2}$ for sufficiently large $n$. The proof of the uniqueness requires a careful use of some sharp estimates for bubbling solutions of mean field equations [24] and a rather involved analysis of suitably defined Pohozaev-type identities as recently developed in [51] in the context of the Chern-Simons-Higgs equations. Moreover, motivated by the Onsager statistical description of two dimensional turbulence, we are bound to obtain a refined version of an estimate about $ρ_n-8πm$ in case the first order evaluated in [24] vanishes.

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arXiv2016

Global bifurcation analysis of mean field equations and the Onsager microcanonical description of two-dimensional turbulence

On strictly starshaped domains of second kind we find natural sufficient conditions which allow the solution of two long standing open problems closely related to the mean field equation $\prl$ below. On one side we catch the global behaviour of the Entropy for the mean field Microcanonical Variational Principle ((MVP) for short) arising in the Onsager description of two-dimensional turbulence. This is the completion of well known results first established in Caglioti et al. Comm.Math.Phys. (1995). Among other things we find a full unbounded interval of strict convexity of the Entropy. On the other side, to achieve this goal, we have to provide a detailed qualitative description of the global branch of solutions of $\prl$ emanating from $\lm=0$ and crossing $\lm=8π$. This is the completion of well known results first established in Suzuki A.I.H.P. (1992) and Chang et al. New Stud. Adv. Math. (2003) for $\lm\leq 8π$, and it has an independent mathematical interest, since global branches of semilinear elliptic equations, with very few well known exceptions, are poorly understood. The (MVP) suggests the right variable (which is the energy) to be used to obtain a global parametrization of solutions of $\prl$. A crucial spectral simplification is obtained by using the fact that, by definition, solutions of the (MVP) maximize the entropy at fixed energy and total vorticity.

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arXiv2016

On a singular Liouville-type equation and the Alexandrov isoperimetric inequality

We obtain a generalized version of an inequality, first derived by C. Bandle in the analytic setting, for weak subsolutions of a singular Liouville-type equation. As an application we obtain a new proof of the Alexandrov isoperimetric inequality on singular abstract surfaces. Interestingly enough, motivated by this geometric problem, we obtain a seemingly new characterization of local metrics on Alexandrov's surfaces of bounded curvature. At least to our knowledge, the characterization of the equality case in the isoperimetric inequality in such a weak framework is new as well.

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arXiv2015

A global existence result for a Keller-Segel type system with supercritical initial data

We consider a parabolic-elliptic Keller-Segel type system, which is related to a simplified model of chemotaxis. Concerning the maximal range of existence of solutions, there are essentially two kinds of results: either global existence in time for general subcritical ($\|ρ_0\|_1<8π$) initial data, or blow--up in finite time for suitably chosen supercritical ($\|ρ_0\|_1>8π$) initial data with concentration around finitely many points. As a matter of fact there are no results claiming the existence of global solutions in the supercritical case. We solve this problem here and prove that, for a particular set of initial data which share large supercritical masses, the corresponding solution is global and uniformly bounded.

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arXiv2014

Self gravitating cosmic strings and the Alexandrov's inequality for Liouville-type equations

Motivated by the study of self gravitating cosmic strings, we pursue the well known method by C. Bandle to obtain a weak version of the classical Alexandrov's isoperimetric inequality. In fact we derive some quantitative estimates for weak subsolutions of a Liouville-type equation with conical singularities. Actually we succeed in generalizing previously known results, including Bol's inequality and pointwise estimates, to the case where the solutions solve the equation just in the sense of distributions. Next, we derive some \uv{new} pointwise estimates suitable to be applied to a class of singular cosmic string equations. Finally, interestingly enough, we apply these results to establish a minimal mass property for solutions of the cosmic string equation which are \uv{supersolutions} of the singular Liouville-type equation.

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arXiv2013

Supercritical Mean Field Equations on convex domains and the Onsager's statistical description of two-dimensional turbulence

We are motivated by the study of the Microcanonical Variational Principle within the Onsager's description of two-dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and "thin" enough domains in the supercritical (with respect to the Moser-Trudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known \un{only} on multiply connected domains. Then we study the structure of these solutions by the analysis of their linearized problems and also obtain a new uniqueness result for solutions of the Mean Field Equation on thin domains whose energy is uniformly bounded from above. Finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and use it together with all the results obtained so far to solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is eventually shown to be concave.

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arXiv2013

Asymptotic analysis of solutions to a gauged O(3) sigma model

We analyze an elliptic equation arising in the study of the gauged O(3) sigma model with the Chern-Simons term. In this paper, we study the asymptotic behavior of solutions and apply it to prove the uniqueness of stable solutions. However, one of the features of this nonlinear equation is the existence of stable nontopological solutions in $\RN$, which implies the possibility that a stable solution which blows up at a vortex point exists. To exclude this kind of blow up behavior is one of the main difficulties which we have to overcome.

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arXiv2012

Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter

We consider the mean field equation: (1) Δu+ρ\frac{e^u}{\int_Ωe^u}=0 & \hbox{in} \;Ω, u=0 & \hbox{on}\;\partialΩ, where $Ω\subset \mathbb{R}^2$ is an open and bounded domain of class $C^1$. In his 1992 paper, Suzuki proved that if $Ω$ is a simply-connected domain, then equation (1) admits a unique solution for $ρ\in[0,8π)$. This result for $Ω$ a simply-connected domain has been extended to the case $ρ=8π$ by Chang, Chen and the second author. However, the uniqueness result for $Ω$ a multiply-connected domain has remained a long standing open problem which we solve positively here for $ρ\in[0,8π]$. To obtain this result we need a new version of the classical Bol's inequality suitable to be applied on multiply-connected domains. Our second main concern is the existence of solutions for (1) when $ρ=8π$. We a obtain necessary and sufficient condition for the solvability of the mean field equation at $ρ=8π$ which is expressed in terms of the Robin's function $γ$ for $Ω$. For example, if equation (1) has no solution at $ρ=8π$, then $γ$ has a unique nondegenerate maximum point. As a by product of our results we solve the long-standing open problem of the equivalence of canonical and microcanonical ensembles in the Onsager's statistical description of two-dimensional turbulence on multiply-connected domains.

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arXiv2012

An improved geometric inequality via vanishing moments, with applications to singular Liouville equations

We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.

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