无穷Laplace方程解的边界正则性

项目名称： 无穷Laplace方程解的边界正则性

项目编号： No.11301411

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 洪广浩

作者单位： 西安交通大学

项目金额： 22万元

中文摘要： 本项目研究齐次和非齐次无穷Laplace方程的粘性解的边界正则性问题。无穷Laplace方程是一个高度退化并且高度非线性的二阶椭圆方程。它是最优Lipschitz延拓变分问题的Euler-Lagrange方程，也可以看做是p-Laplace方程当p趋于无穷时的极限方程，并且与概率论中的拔河比赛问题相关。该方程在图像处理和最优输运问题中也有重要应用。近几年无穷Laplace方程正逐渐成为椭圆与抛物方程领域的一个研究热点，其中边界正则性方面的研究刚刚开展。本项目拟研究无穷Laplace方程边界正则性方面的三个问题：（1）非齐次方程在可微边界条件下解的边界逐点可微性；（2）一阶连续可微边界条件下解的边界连续可微性；（3）凸区域和Reifenberg平坦区域上局部零边值问题的解的边界可微性。旨在探索解的边界逐点可微性与连续可微性成立的最佳边界条件。

中文关键词： 无穷Laplace方程；粘性解；边界正则性；；

英文摘要： We study the boundary regularity of the viscosity solutions of infinity Laplace equation and inhomogeneous infinity Laplace equation in this program. Infinity Laplace equation is a highly degenerate and highly nonlinear elliptic equation of second order. It is the Euler-Lagrange equation for the optimal Lipschitz extension variational problem and can be also seen as the limit equation of p-Laplace equations as p go to infinity. It is also related to the tug of war game in probility theory. Recently, this equation has found important applications in image process and optimal mass transportation problems. The theory of infinity Laplace equation is becoming a hot topic in the field of elliptic and parabolic equations. Research has only recently begun on the boundary regularity. In this program we propose three problems on the boundary regularity of the solutions of infinity Laplace equations: (i)the boundary differentiability of solutions to the inhomogeneous infinity Laplace equation with the assumptions that the boundary of domain and the given Dirichlet boundary data are differentiable on the boundary; (ii)the boundary continuous differentiability of solutions to the infinity Laplace equation with the C^1 assumptions on both the boundary of domain and the given Dirichlet boundary data; (iii)the boundary differen

英文关键词： infinity Laplace equation；viscosity solution；boundary regularity；；