Schedule-at-a-Glance

15 PRIMA 2013 AbstractsIn this talk, we study the weak continuity of k-curvaturemeasures of locally bounded, upper semicontinous functionswhich are subharmonic with respect to k-curvatureoperators. The the proof uses the monotonicity integralinequality. This is a joint work with Qiuyi Dai and Xu-jiaWang.Special Session 7Geometric Aspects of Semilinear EllipticEquations: Recent Advances & Future PerspectivesRegularization of point vortex solution forEuler equationDaomin CaoChinese Academy of Sciences, Chinadmcao@amt.ac.cnIn this talk the speaker will talk about the regularizationof point vortex solution for Euler equation of 2 dimension.The existence of point vortex solutions is closelyrelated to the so called Kirchhoff Routh function. Thecritical points of Kirchhoff Routh function can deduce asteady point vortex solution. For a given critical point ofKirchhoff Routh function, can one establish the existenceof smooth steady solutions approximating the point vortexsolution deduced by the point? He will talk about theprevious study concerning this respect. A newly obtainedresults by the speaker and his collaborators, Liu ZhongYuan and Wei Juncheng, will be introduced.Nonlocal minimal surfacesJuan DávilaUniversidad de Chile, Chilejdavila@dim.uchile.clCaffarelli, Roquejoffre and Savin (2010) introduced a notionof nonlocal minimal surface, which is a boundary ofa set that is minimal with respect to an s-perimeter functional,where 0 < s < 1. It is natural to consider criticalpoints of the s-perimeter, which we call s-minimal surfaces.Up to now there are very few examples of such sets.Starting from the property that as s → 1, the s-minimalequation approaches the classical minimal surface equation,we construct new examples of s-minimal surfaces fors close to 1.This is joint work with Manuel del Pino (Universidad deChile) and Juncheng Wei (Chinese University Of HongKong and University of British Columbia).Solutions for a semilinear elliptic equation indimension two with supercritical growthIgnacio GuerraUniversidad de Santiago de Chile, Chileignacio.guerra@usach.clWe consider the problem−∆u = λue up , u > 0, in Ω,u = 0 on ∂Ω,where Ω ⊂ R 2 and p > 2. Let λ 1 be the first eigenvalueof the Laplacian. For each λ ∈ (0, λ 1 ), we prove the existenceof solutions for p sufficiently close to 2. In the caseof Ω a ball, we also describe numerically the bifurcationdiagram (λ, u) for p > 2.On the entire radial solutions arising inChern-Simons equationsHsin-Yuan HuangNational Sun Yat-sen University, Taiwan, R.O.Chyhuang@math.nsysu.edu.twIn this paper, we study the the entire radial solutions of(∗):∆u = − (1 + a)(e u − (1 + a)e 2u + ae u+v )+ a(e v − (1 + b)e 2v + be u+v )+ 4πN 1 δ 0∆v = − (1 + b)(e v − (1 + b)e 2v + be u+v )+ b(e u − (1 + a)e 2u + ae u+v )+ 4πN 2 δ 0 .Here, a > 0, b > 0, N 1 ≥ 0, and N 2 ≥ 0. This systemis motivated by studying the equations of relativisticnon-Abelian Chern-Simons model proposed by by Kao-Lee and Dunne and Gudnason model of N =2 supersymmetricYang-Mills-Cherns-Simons-Higgs theory. Understandingthe structure of entire radial solutions is oneof fundamental issues for the system of nonlinear equations.Under certain technical conditions for a and b,we prove that any entire radial solutions of (∗) must beone of topological, non-topological and mixed type solutions,and completely classify the asymptotic behaviorsat infinity of these solutions. As an application of thisclassification, we prove that the two components u and vhave intersection at most finite times.Existence and non-existence of positive solutionson the Hardy-Littlewood-Sobolev typesystemsCongming LiShanghai Jiao Tong University, China and University ofColorado at Boulder, USAcli@colorado.eduIn this presentation, we provide some existence, nonexistence,and classification of positive solutions for Hardy-Littlewood-Sobolev type systems. In deriving these results,some useful methods are created. We will brieflyintroduce some of the involved methods.Singly periodic solutions of the Allen-Cahnequation and the Toda latticeYong LiuNorth China Electric Power University, Chinaliuyong@ncepu.edu.cnWe study the Allen-Cahn equation in the plane:−∆u = u − u 3 , in R 2 .Of interest are bounded entire solutions whose behaviorat infinity is in some sense simple. Examples arethose with finite Morse index, which have been extensivelystudied in the past. Here we will construct solutionswhich are periodic in one direction in the plane, butare not doubly periodic. The core of the construction isthe one-soliton solution of the Toda lattice, which is aclassical integrable system. This is joint work with M.Kowalczyk, F. Pacard and J. Wei.Critical Trudinger Moser equation in R 2Monica MussoUniversidad Católica de Chile, Chilemmusso@mat.puc.cl