Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics

Motivated by the mean field equations with probability measure derived by Sawada-Suzuki and by Neri in the context of the statistical mechanics description of two-dimensional turbulence, we study the semilinear elliptic equation with probability measure: {equation*} -\Delta v=\lambda\int_I V(\alpha,x,v)e^{\alpha v}\,\Pda -\frac{\lambda}{|\Omega|}\iint_{I\times\Om}V(\alpha,x,v)e^{\alpha v}\,\Pda dx, {equation*} defined on a compact Riemannian surface. This equation includes the above mentioned equations of physical interest as special cases. For such an equation we study the blow-up properties of solution sequences. The optimal Trudinger-Moser inequality is also considered

Mass quantization and minimax solutions for Neri's mean field equation in 2D-turbulence

We study the mean field equation derived by Neri in the context of the statistical mechanics description of 2D-turbulence, under a ``stochastic" assumption on the vortex circulations. The corresponding mathematical problem is a nonlocal semilinear elliptic equation with exponential type nonlinearity, containing a probability measure $mathcal Pinmathcal M([-1,1])$ which describes the distribution of the vortex circulations. Unlike the more investigated ``deterministic" version, we prove that Neri's equation may be viewed as a perturbation of the widely analyzed standard mean field equation, obtained by taking $mathcal P=delta_1$. In particular, in the physically relevant case where $mathcal P$ is non-negatively supported and $mathcal P({1})>0$, we prove the mass quantization for blow-up sequences. We apply this result to construct minimax type solutions on bounded domains in $mathbb R^2$ and on compact 2-manifolds without boundary

On the blowup of solutions to Liouville type equations.

We estimate some complex structures related to perturbed Liouville equations defined on a compact Riemannian 2-manifold. As a byproduct, we obtain a quick proof of the mass quantization and we locate the blow-up points

An Obstacle Problem for Noncoercive Operators

We study the obstacle problem for second order nonlinear equations whose model appears in the stationary diffusion-convection problem. We assume that the growth coefficient of the convection term lies in the Marcinkiewicz spaceweak-LN

Noncoercive quasilinear elliptic operators with singular lower order terms

We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular

Foreign children with cancer in Italy

<p>Abstract</p> <p>Background</p> <p>There has been a noticeable annual increase in the number of children coming to Italy for medical treatment, just like it has happened in the rest of the European Union. In Italy, the assistance to children suffering from cancer is assured by the current network of 54 centres members of the Italian Association of Paediatric Haematology and Oncology (AIEOP), which has kept records of all demographic and clinical data in the database of Mod.1.01 Registry since 1989.</p> <p>Methods</p> <p>We used the information stored in the already mentioned database to assess the impact of immigration of foreign children with cancer on centres' activity, with the scope of drawing a map of the assistance to these cases.</p> <p>Results</p> <p>Out of 14,738 cases recorded by all centres in the period from 1999 to 2008, 92.2% were born and resident in Italy, 4.1% (608) were born abroad and living abroad and 3.7% (538) were born abroad and living in Italy. Foreign children cases have increased over the years from 2.5% in 1999 to. 8.1% in 2008.</p> <p>Most immigrant children came from Europe (65.7%), whereas patients who came from America, Asia and Oceania amounted to 13.2%, 10.1%, 0.2%, respectively. The immigrant survival rate was lower compared to that of children who were born in Italy. This is especially true for acute lymphoblastic leukaemia patients entered an AIEOP protocol, who showed a 10-years survival rate of 71.0% vs. 80.7% (p < 0.001) for immigrants and patients born in Italy, respectively.</p> <p>Conclusions</p> <p>Children and adolescents are an increasingly important part of the immigration phenomenon, which occurs in many parts of the world. In Italy the vast majority of children affected by malignancies are treated in AIEOP centres. Since immigrant children are predominantly treated in northern Italy, these centres have developed a special expertise in treating immigrant patients, which is certainly very useful for the entire AIEOP network.</p