无穷Laplace方程解的边界正则性

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 differentiability of solutions to the infinity Laplace equation in convex domains and Reifenbergflat domains with partly vanishing Dirichlet boundary data. Our goal in this program is to discover the optimal boundary conditions to boundary differentiability or boundary continuous differentiability of the solutions to infinity Laplace equations.

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。

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。具体来讲我们考虑如下三个问题。.（i）非齐次方程在可微边界条件下解的边界可微性。.（ii）一阶连续可微边界条件下，解的边界连续可微性。.（iii）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。.我们得到的主要研究结果如下。.（i）我们证明了在可微边界条件下非齐次无穷Laplace方程的粘性解在边界是可微的。.（ii）我们构造了一个反例表明在连续可微的区域上无穷调和函数在边界不一定是连续可微的，即使在2维且边值函数光滑。这负面地回答了Wang-Yu文章中的一个问题。.（iii）我们研究了无穷调和函数在凸区域角点的斜率和可微性成立的条件。我们证明了函数的斜率不超过边值函数的梯度的模。我们证明了当边值函数的梯度方向或负梯度方向指向区域内部时，函数在这个边界角点是可微的；当边值函数的梯度方向和负梯度方向都不指向区域内部时，函数在这个边界角点是可微的当且仅当函数的斜率等于边值函数的梯度的模。