2010 Putnam A3

Suppose that the function $h:\mathbb{R}^2\to \mathbb{R}$ has continuous partial

derivatives and satisfies the equation

$$

h(x,y) = a \frac{\partial h}{\partial x}(x,y) +

b \frac{\partial h}{\partial y}(x,y)

$$

for some constants $a,b$. Prove that if there is a constant $M$ such that

$|h(x,y)|\leq M$ for all $(x,y) \in \mathbb{R}^2$, then $h$ is identically zero.

假设函数 $h:\mathbb{R}^2\to \mathbb{R}$ 具有连续部分

导数并满足方程

$$

h(x,y) = a \frac{\partial h}{\partial x}(x,y) +

b \frac{\partial h}{\partial y}(x,y)

$$

对于一些常数$a,b$。证明如果存在常数 $M$ 使得

$|h(x,y)|\leq M$ 对于所有 $(x,y) \in \mathbb{R}^2$，则 $h$ 相同为零。