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.Comment: some references adde

New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces

We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.Comment: to appear in GAF

A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$-\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$},$$ and some of its variants and extensions.Comment: 39 page

The role of Liouville type systems in the study of non-topological Chern-Simons vortices

The International Congress on Mathematical Physics is a major conference in its field that attracts a very wide spectrum of researchers. Held every three years, it provides an overview of recent developments and achievements in mathematical physics. This volume presents the plenary lectures and invited topical session lectures from the XVIIth ICMP, which was held in Aalborg, Denmark, August 2012. It also includes additional material from the Congress. In this volume, one can find survey lectures on orthogonal polynomials, random systems, information theory in physics, several aspects of quantum field theory and quantum mechanics, general relativity, and classical and quantum dynamical systems

The role of Liouville type systems in the study of non-topological Chern-Simons vortices

The International Congress on Mathematical Physics is a major conference in its field that attracts a very wide spectrum of researchers. Held every three years, it provides an overview of recent developments and achievements in mathematical physics. This volume presents the plenary lectures and invited topical session lectures from the XVIIth ICMP, which was held in Aalborg, Denmark, August 2012. It also includes additional material from the Congress. In this volume, one can find survey lectures on orthogonal polynomials, random systems, information theory in physics, several aspects of quantum field theory and quantum mechanics, general relativity, and classical and quantum dynamical systems

Cutaneous manifestations in a series of 417 patients with SARS-CoV-2 infection: epidemiological and clinical correlates of chilblain like lesions

none8A variety of dermatological lesions have been described in COVID-19, although the prevalence and pathogenic relationship remain unclear particularly for chilblain-like lesions. Dermatological examination was performed in a prospective cohort of consecutive patients seen at the service for SARS-CoV-2 infection. Out of 417 patients with confirmed SARS-CoV-2 infection [median age 29.5 years (range 15-65); 62.5% males], dermatological lesions were detected in 7 (1.7%). Three patients had acral lesions; their age (range) was 15-29 years; all had a negative nasopharyngeal swab and developed IgG and/or IgM-specific antibodies; all presented none or mild symptoms. A fourth patient remained negative at repeated testing; mother, father and sister had a documented mild COVID-19. Non-acral lesions were observed in four older patients, with severe COVID-19. Chilblain-like lesions may be the sole manifestation of SARS-CoV-2 infection; their presence in asymptomatic school children and adolescents should be considered a potential signal of familial or community spread of the virus.noneBrancaccio, G; Gussetti, N; Sasset, L; Alaibac, M; Tarantello, M; Salmaso, R; Trevenzoli, M; Cattelan, A MBrancaccio, G; Gussetti, N; Sasset, L; Alaibac, M; Tarantello, M; Salmaso, R; Trevenzoli, M; Cattelan, A