Can the following norm be put on $ C^1(D)$ that can make it into a Banach space?

Question

I know that $C^1([a,b])$ with norm $\|u\|=\|u\|_\infty+\|u'\|_\infty $ is a Banach space. My question is: Can the following norm be put on $ C^1(D), D \subset \mathbb{R}^2$ that can make it into a Banach space? $$\|u\|_1=\|u\|_\infty+\max\left( \left| \frac{\partial u}{\partial x_1} \right|, \left| \frac{\partial u}{\partial x_2} \right| \right).$$