设z=f(x,y)在有节闭区域D上具有二阶连续偏导,且d^2Z/dx^2+d^2Z/dy^2=0,d^2Z/dxdy≠0.证明;z的最值在D的边界上取得

Question

设z=f(x,y)在有节闭区域D上具有二阶连续偏导,且d^2Z/dx^2+d^2Z/dy^2=0,d^2Z/dxdy≠0.证明;z的最值在D的边界上取得

Answer 1

反证法：若z＝f(x,y)在D的内部达到最值,不妨设
(x0,y0)点是最大值,内部的最值必是极值,于是由
极值的必要条件知道有az/ax＝az/ay＝0,
且Hessian阵（a^2z/ax^2 a^2z/axay
a^2z/axay a^2z/ay^2）
是半负定阵,即必有
a^2z/ax^2