Weak Derivatives

Jose27 · #1 $Old$ June 8th, 2009, 07:41 PM

If $\Omega \subset \mathbb{R} ^n$ open and connected , $u \in L_{loc}^1 ( \Omega) = \{ u: \Omega \longrightarrow \mathbb{R}$ : $u$ integrable in $\Omega _1$ for every open $\Omega _1 \hspace{2 mm} such \hspace{2 mm} that \hspace{2 mm} \overline{ \Omega_1} \subset \Omega \hspace{2 mm} is \hspace{2 mm} compact \hspace{2 mm} in \hspace{2 mm} \Omega \}$ is weakly differentiable in $\Omega$ and $D_{i} u =0$ $\forall i=1, ... , n$ then $u$ is constant a.e. in $\Omega$

I don't even know where to begin with this one, the usual proof for when $u$ is differentiable does not apply since we don't really have a mean value theorem for this derivatives, so... I'm stuck. Any help is appreciated.

putnam120 · #2 $Old$ June 8th, 2009, 07:56 PM

What does $D_iu$ mean? this seems like an interesting problem and would like to try and help.

Jose27 · #3 $Old$ June 8th, 2009, 08:12 PM

Quote:

Originally Posted by putnam120 $View Post$

What does $D_iu$ mean? this seems like an interesting problem and would like to try and help.

$D_{i} u$ is the $i$ -eth (spell?) weak derivative of $u$ ie. it is a function $v_i \in L_{loc} ^1 ( \Omega)$ such that $\int_{ \Omega} u \frac{ \partial \phi}{ \partial x_i } = - \int_{ \Omega} v_{i} \phi$ for all $\phi \in C_{c} ^{\infty} ( \Omega)$ (the space of all infinitely differentiable function to $\mathbb{R}$ with compact ( in $\Omega$ ) support).

The weak derivatives satisfy the common propierties of the partial derivatives, ie, it's unique(modulo the relation of being equal a.e.), the weak derivative of a sum is the sum of the derivatives, etc.