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Paolo Cosentino

Publications and source records attributed to Paolo Cosentino.

6 recordsLinked to original sources
arXiv2026

Blow up and Concentration without Quantization: sharp Harnack type inequalities

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we refine the blow up analysis for sequences of solutions of a class of perturbed singular Liouville equations which share the phenomenon of "blow up and concentration without quantization". The problem is delicate because we are dealing with the exact threshold value above which one meets the well known "concentration without quantization" phenomenon, as recently pushed forward in [C.S. Lin, G. Tarantello, C. R. Math. Acad. Sci. Paris (2016)] and in [Y. Lee, C.S. Lin, G. Tarantello, W. Yang, Comm. PDE. (2017)]. First of all we need a new sharp Harnack type inequality for this particularly rich singular limit. However this is not enough, since the growth of the conformal factor inherited by the singularity prevents the use of classical quantization arguments. We solve also this issue with different strategies for "fast" and "slow" blow up, by a careful adaptation of arguments based on the Pohozaev identity, elliptic estimates and "Sup+CInf" inequalities in the same spirit of [C.C. Chen, C.S. Lin, Comm. An. Geom. (1998)].

math.AP↗
arXiv2026

Onsager's Mean Field Theory of Vortex Flows with Singular Sources: Blow-Up and Concentration without Quantization

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we make a first step in the generalization of the mean field theory in [Caglioti, Lions, Marchioro, Pulvirenti; Comm. Math. Phys. (1995)]. On one side we prove the equivalence of statistical ensembles, on the other side we are bound to the analysis of a new blow up phenomenon, which we call "blow up and concentration without quantization", where the mass associated with the concentration is allowed to take values in a full interval of real numbers. This singular behavior may be regarded as lying between the classical blow up-concentration-quantization and the blow up without concentration phenomenon first proposed in [Lin, Tarantello; C.R. Math. Acad. Sci. Paris (2016)]. A careful analysis is needed to generalize known pointwise estimates in this non standard context, resulting in a complete description of the allowed asymptotic profiles.

math.AP↗
arXiv2026

A Harnack-type inequality for a perturbed singular Liouville Equation

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we obtain a Harnack-type inequality for sequences of solutions of the following perturbed Liouville equation, \begin{equation}\nonumber -Δv_n=\left({ε_n^2+|x|^2}\right)^{α_n}V_n(x)e^{\displaystyle v_n} \qquad\text{in} \,\,\, Ω, \end{equation} where $ε_n\to0^+$, $α_n\toα_\infty\in(-1,1)$, $Ω$ is a bounded domain in $\mathbb{R}^2$ containing the origin and $V_n$ satisfies, \begin{equation}\nonumber 0<a\leq V_n\leq b<+\infty, \,\, V_n\in C^{0}(Ω), \,\,V_n\to V \,\, \text{locally uniformly in}\,\,Ω. \end{equation}

math.AP↗
arXiv2024

A Harnack type inequality for singular Liouville type equations

We obtain a Harnack type inequality for solutions of the Liouville type equation, \begin{equation}\nonumber -Δu=|x|^{2α}K(x)e^{\displaystyle u} \qquad\text{in} \,\,\, Ω, \end{equation} where $α\in(-1,0)$, $Ω$ is a bounded domain in $\mathbb{R}^2$ and $K$ satisfies, \begin{equation}\nonumber 0<a\leq K(x)\leq b<+\infty. \end{equation} This is a generalization to the singular case of a result by C.C. Chen and C.S. Lin [Comm. An. Geom. 1998], which considered the regular case $α=0$. Part of the argument of Chen-Lin can be adapted to the singular case by means of an isoperimetric inequality for surfaces with conical singularities. However, the case $α\in(-1,0)$ turns out to be more delicate, due to the lack of traslation invariance of the singular problem, which requires a different approach.

math.AP↗
arXiv2023

On the first eigenvalue of Liouville-type problems

The aim of this note is to study the spectrum of a linearized Liouville-type problem, characterizing the case in which the first eigenvalue is zero. Interestingly enough, we obtain also point-wise information on the associated first eigenfunction. To this end, we refine the Alexandrov-Bol inequality suitable for our problem and characterize its equality case.

math.AP↗