The Liouville Equation with Singular Data: A Concentration-Compactness Principle via a Local Representation Formula

AbstractFor a bounded domain Ω⊂R2, we establish a concentration-compactness result for the following class of “singular” Liouville equations:−Δu=eu−4π∑j=1mαjδpj in Ω where pj∈Ω, αj>0 and δpj denotes the Dirac measure with pole at point pj, j=1,…,m. Our result extends Brezis–Merle's theorem (Comm. Partial Differential Equations16 (1991) 1223–1253) concerning solution sequences for the “regular” Liouville equation, where the Dirac measures are replaced by Lp(Ω)-data p>1. In some particular case, we also derive a mass-quantization principle in the same spirit of Li–Shafrir (Indiana Univ. Math. J.43 (1994) 1255–1270). Our analysis was motivated by the study of the Bogomol'nyi equations arising in several self-dual gauge field theories of interest in theoretical physics, such as the Chern–Simons theory (“Self-dual Chern–Simons Theories” Lecture Notes in Physics, Vol. 36, Springer-Verlag, Berlin, 1995) and the Electroweak theory (“Selected Papers on Gauge Theory of Weak and Electromagnetic Interactions,” World Scientific, Singapore)

OPTICAL AND FUNCTIONAL PERFORMANCE OF PHAKIC ANGLE-SUPPORTED INTRAOCULAR LENS FOR THE CORRECTION OF HIGH MYOPIA IN 18-MONTHS FOLLOW-UP STUDY

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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

Comparison of Ophthalmologists versus Dermatologists for the Diagnosis and Management of Periorbital Atypical Pigmented Skin Lesions

: Background/Objectives: Lentigo maligna (LM) and lentigo maligna melanoma (LMM) are significant subtypes of melanoma, with an annual incidence of 1.37 per 100,000 people in the U.S. These skin tumors, often found in photo-exposed areas such as the face, are frequently misdiagnosed, leading to delayed treatment or unnecessary excisions, especially in the elderly. Facial melanocytic skin tumors (lentigo maligna-LM/lentigo maligna melanoma-LMM) and their simulators (solar lentigo, pigmented actinic keratosis, seborrheic keratosis and lichen planus-like keratosis) often affect the periocular region. Thus, their diagnosis and management can involve different medical figures, mainly dermatologists and ophthalmologists. This study aimed to evaluate the ability of ophthalmologists to diagnose and manage pigmented skin lesions of the periorbital area. Methods: A multicentric, retrospective, cross-sectional study on a dataset of 79 periorbital pigmented skin lesions with both clinical and dermoscopic images was selected. The images were reviewed by six ophthalmologists and two dermatologists. Descriptive statistics were carried out, and the accuracy, sensitivity, and specificity, with their 95% confidence interval (95% CI), were estimated. Results: Ophthalmologists achieved a diagnostic accuracy of 63.50% (95% CI: 58.99-67.85%), while dermatologists achieved 66.50% (95% CI: 58.5-73.8). The sensitivity was lower for ophthalmologists in respect to dermatologists, 33.3% vs. 46.9%, respectively. Concerning the case difficulty rating, ophthalmologists rated as "difficult" 84% of cases, while for dermatologists, it was about 30%. Management was also consistently different, with a "biopsy" decision being suggested in 25.5% of malignant lesions by ophthalmologists compared with 50% of dermatologists. Conclusions: Ophthalmologists revealed a good diagnostic potential in the identification of periorbital LMs/LMMs. Given progressive population ageing and the parallel increase in facial/periorbital skin tumors, the opportunity to train new generations of ophthalmologists in the early diagnosis of these neoformations should be considered in the next future, also taking into account the surgical difficulty/complexity of this peculiar facial area