带间断非线性项椭圆型偏微分方程

本项目主要研究具有重要物理背景的非线性偏微分方程，如研究描述不可压理想流体的Euler方程定常点涡解的非线性自由边值问题和与环形腔体（托克马克装置）中等离子的平衡态相关模型所导出的非线性椭圆型方程。这些方程具有间断非线性项，以往的研究很少涉及。撇开这些问题的物理和应用背景，对这些方程的研究需要发展新的工具，其本身也是有意义的。我们将研究当某个参数很小时解集中在一个或几个点的存在性，从而给出定常点涡解的光滑逼近解。我们还将研究非线性Schrodinger方程驻波解的轨道稳定性等。我们的研究将丰富变分学和偏微分方程的现有理论。

In this project we mainly invistigate those nonlinear elliptic equations which have important physical background.For instance,we will study the nonlinear free boundary problems related to steady vortex solution of Euler equations describing ideal incompressible fluid and the nonlinear elliptic equations derived from model related to equilibrium of a plasma confined in a toroidal cavity(a Tokamak machine). The nonlinear terms of those equations have the discontinuity and were rarely studied in the past. Apart from their physical and application background, the study of those equations itself is of great interest since new techniques are needed to deal with such nonlinearities. We will study the existence of solutions concentrating at one or a couple of points when the parameter is small, which will derives the existence of classic solution approximating steady point vortex solutions.We will also study the orbit stability of standy waves of nonlinear Schrodinger equations. Our research will enrich the existing theory of calculus of variation and partial differential equations.