Let be $u\in W^{m,p}(\mathbb{R^n})$. Show that if a one partial derivate of $u$ is zero, then $u = 0$.
How I can be able to start this proof?
Let be $u\in W^{m,p}(\mathbb{R^n})$. Show that if a one partial derivate of $u$ is zero, then $u = 0$.
How I can be able to start this proof?
You may assume that $\frac{\partial u}{\partial x_n} = 0$, so that $u(x_1,\ldots, x_{n-1},x_n) = u_0(x_1, \ldots, x_{n-1})$ with some function $u_0$. Now $\|u\|_p^p = \int_{\mathbb{R}^n} |u|^p \, dx_1 \cdots dx_n = \int_{\mathbb{R}} \|u_0\|_p^p \, dx_n, $ using Fubini and integrating out all variables except $x_n$. The only way that the integral of a constant over $\mathbb{R}$ can be finite is if that constant is $0$, so you get $\|u_0\|_p = 0$, and so $\|u\|_p = 0$, and $u=0$ almost everywhere.