平面上二阶椭圆方程的边界唯一延拓性质(英文)

A BOUNDARY UNIQUE CONTINUATION PROPERTY FOR THE SOLUTIONS OF ELLIPTIC EQUATIONS IN THE PLANE

设w（r）非负非增，δ＞0，D为平面上的区域，z0∈D，u是D上的一个二阶椭圆方程的解，且|u(z)|≤Cexp{-w(|w(|z-z0|)}(A)我们证明了，如果，则u=0；若，则有上半平面上的有界调和函数u使（A）成立．

Let w(r) be a nonnegative,decrease function for r > 0. It is shown that if ,u is a solution of second order elliptic equation on a smooth domain D in the plane,and if at a boundary point x0 of D, |u(z)|≤ C exp { - w(|z - z0|)} for z∈D,then u ≡0 in D. On the orther hand,if ,then there exists a nonzero bounded harmonic function u(z) in R2+ such that |u(z)|≤ Cexp { - w(|z|) }.