Problem

請寫出計算及證明過程，勿只寫答案。

5. Let $U$ be the region in $\mathbb{R}^2$ defined by $r\le 1+\cos\theta$ in polar coordinate. Let $\gamma$ be the boundary of $U$; namely, $\gamma$ is given by $r=1+\cos\theta$ in polar coordinate. Does there exist a second continuously differentiable function $f(x,y)$ on $\mathbb{R}^2$ such that

$$\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}=2x+3y^2$$

at every $(x,y)\in U$, and

$$\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)=\left(4x^2-3y^2+x,\ y^3-6y\right)$$

at every $(x,y)\in \gamma$? If your answer is YES, construct such a function. If your answer is NO, give a proof.