Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Topology and Geometry

Infinite Dimensional Morse Theory and Multiple Solution Problems (K. C. Chang)

We study the problem of prescribing the Gaussian curvature on surfaces with conical singularities in supercritical regimes. Using a Morse-theoretical approach, we prove a general existence theorem on surfaces with positive genus, with a generic multiplicity result.

Supercritical conformal metrics on surfaces with conical singularities

We consider the following mean field equation: 

Δgv+ρ(h*ev∫Mh*ev−1)=4π∑j=1Nαj(δqj−1) on M,

where M is a compact Riemann surface with volume 1, h* is a positive C1 function on M, and ρ and αj are constants satisfying αj > −1. In this paper, we derive the topological-degree-counting formula for noncritical values of ρ. We also give several applications of this formula, including existence of the curvature  + 1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings. © 2015 Wiley Periodicals, Inc.

Mean Field Equation of Liouville Type with Singular Data: Topological Degree

https://www.sissa.it/fa/download/publications/JFA-DeMarchis.pdf

Generic multiplicity for a scalar field equation on compact surfaces

We consider the semilinear Lane-Emden problem 1uDjuj p 1 u in; uD 0 on@; (Ep) where p > 1 andis a smooth bounded domain of R 2 . The aim of the paper is to analyze the asymptotic behavior of sign-changing solutions of (Ep) as p! 1. Among other results we show, under some symmetry assumptions on , that the positive and negative parts of a family of symmetric solutions concentrate at the same point as p ! 1, and the limit profile looks like a tower of two bubbles given by superposition of a regular and a singular solution of the Liouville problem in R 2 .

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=17&iss=8&rank=4

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

We obtain sufficient conditions for the existence of the Ambjorn-Olesen [“On electroweak magnetism,” Nucl. Phys. B315, 606–614 (1989)10.1016/0550-3213(89)90004-7] electroweak N-vortices in case N ⩾ 1 and therefore generalize earlier results [D. Bartolucci and G. Tarantello, “Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory,” Commun. Math. Phys. 229, 3–47 (2002)10.1007/s002200200664; J. Spruck and Y. Yang, “On multivortices in the electroweak theory I: Existence of periodic solutions,” Commun. Math. Phys. 144, 1–16 (1992)10.1007/BF02099188] which handled the cases N ∈ {1, 2, 3, 4}. The variational argument provided here has its own independent interest as it generalizes the one adopted by Ding et al. [“Existence results for mean field equations,” Ann. Inst. Henri Poincare, Anal. Non Lineaire 16, 653–666 (1999)10.1016/S0294-1449(99)80031-6] to obtain solutions for Liouville-type equations on closed 2-manifolds. In fact, we obtain at once...

On the Ambjorn-Olesen electroweak condensates

We are motivated by the study of the Microcanonical Variational Principle within 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 only on multiply connected domains. We then study the structure of these solutions by the analysis of their linearized problems and we 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, combining it with all the results obtained so far, we solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is shown to be concave.

Francesca De Marchis

Papers

Supercritical conformal metrics on surfaces with conical singularities

Generic multiplicity for a scalar field equation on compact surfaces

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

On the Ambjorn-Olesen electroweak condensates

Supercritical Mean Field Equations on Convex Domains and the Onsager’s Statistical Description of Two-Dimensional Turbulence