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Thread: Weak Derivatives

  1. #1
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    Weak Derivatives

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

    I don't even know where to begin with this one, the usual proof for when $\displaystyle u$ 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.
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  2. #2
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    What does $\displaystyle D_iu$ mean? this seems like an interesting problem and would like to try and help.
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  3. #3
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    Quote Originally Posted by putnam120 View Post
    What does $\displaystyle D_iu$ mean? this seems like an interesting problem and would like to try and help.
    $\displaystyle D_{i} u$ is the $\displaystyle i$-eth (spell?) weak derivative of $\displaystyle u$ ie. it is a function $\displaystyle v_i \in L_{loc} ^1 ( \Omega)$ such that $\displaystyle \int_{ \Omega} u \frac{ \partial \phi}{ \partial x_i } = - \int_{ \Omega} v_{i} \phi$ for all $\displaystyle \phi \in C_{c} ^{\infty} ( \Omega)$ (the space of all infinitely differentiable function to $\displaystyle \mathbb{R}$ with compact ( in $\displaystyle \Omega$ ) 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.
    Last edited by Jose27; Jun 8th 2009 at 06:33 PM.
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  4. #4
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    Quote Originally Posted by Jose27 View Post
    If $\displaystyle \Omega \subset \mathbb{R} ^n$ open and connected , $\displaystyle u \in L_{loc}^1 ( \Omega) = \{ u: \Omega \longrightarrow \mathbb{R}$ : $\displaystyle u$ integrable in $\displaystyle \Omega _1$ for every open $\displaystyle \Omega _1 \hspace{2 mm} such \hspace{2 mm} that \hspace{2 mm} \overline{ \Omega_1} \subset \Omega \hspace{2 mm} is \hspace{2 mm} compact \hspace{2 mm} in \hspace{2 mm} \Omega \}$ is weakly differentiable in $\displaystyle \Omega $and $\displaystyle D_{i} u =0$ $\displaystyle \forall i=1, ... , n$ then $\displaystyle u$ is constant a.e. in $\displaystyle \Omega$

    I don't even know where to begin with this one, the usual proof for when $\displaystyle u$ 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.
    Okay, it's been a while since I posted this but I have an answer now so if anyone's interested here it is:

    We're going to prove a generalization of the result, namely: Let $\displaystyle u\in W_{loc} ^{k,p} (\Omega ) := \{ u\in L_{loc}^p(\Omega ) : D^{\alpha } u \in L_{loc}^p (\Omega) \ \mbox{for all} \ |\alpha| \leq k \}$ (where $\displaystyle \alpha =(\alpha_1,...,\alpha _n) \in \mathbb{N} ^n$ and $\displaystyle D^{\alpha } u$ satisfies $\displaystyle \int _V uD^{\alpha }\phi =(-1)^{| \alpha |}\int_V D^{\alpha }u\phi$ for all $\displaystyle \phi \in C_0^{\infty } (V)$ and $\displaystyle V \subset \subset \Omega$ ) and $\displaystyle D^{\alpha } u =0$ for all $\displaystyle |\alpha | =k$ then $\displaystyle u \in \mathbb{R} [x_1,...,x_n]$ and $\displaystyle \deg (u) \leq k-1$.

    Let $\displaystyle \eta : \mathbb{R} ^n \rightarrow \mathbb{R}$ where $\displaystyle \eta (x) = ce^{\frac{1}{\| x \| ^2 -1}}$ where $\displaystyle c = \int_{\mathbb{R} ^n} \eta$ and let $\displaystyle \eta _{\varepsilon } (x)= \varepsilon ^{-n} \eta \left( \frac{x}{\varepsilon } \right)$. Define $\displaystyle u_{\varepsilon } = u \ast \eta _{\varepsilon }$ then $\displaystyle u_{\varepsilon } \in C^{\infty } (\Omega _{\varepsilon } )$ where $\displaystyle \Omega _{\varepsilon } = \{ x\in \Omega : d(x, \partial \Omega ) > \epsilon \}$ and $\displaystyle u_{\varepsilon } \rightarrow u$ in $\displaystyle L_{loc} ^p ( \Omega )$ when $\displaystyle \varepsilon \rightarrow 0$, and even more: $\displaystyle D^{\alpha } (u_{\varepsilon }) = (D^{\alpha } u)_{\varepsilon }$ where the derivative in the right hand side is the weak derivative (this identity follows at once from the identity $\displaystyle D_x^{\alpha } \eta_{\varepsilon } (x-y) = (-1)^{|\alpha |}D_y^{\alpha } \eta _{\varepsilon } (x-y)$ and the definition of weak derivative).

    The argument is inductive, so we start with the case $\displaystyle k=1$:

    Let $\displaystyle V \subset \subset \Omega$ and $\displaystyle \varepsilon >0$ be small enough such that $\displaystyle \overline{V} \subset \Omega _{\varepsilon }$ then $\displaystyle u_{\varepsilon } \in W^{1,p} (V)$ and we have that $\displaystyle (D^iu)_{\varepsilon } = D^i(u_{ \varepsilon })$ and so $\displaystyle D^i(u_{\varepsilon}) =0$ for all $\displaystyle i=1,...,n$ which implies that $\displaystyle u_{\varepsilon }$ is constant and so $\displaystyle u=c$ in $\displaystyle V$ (since $\displaystyle u_{\varepsilon }$$\displaystyle \rightarrow u$ in $\displaystyle L^p (V)$). Since $\displaystyle \Omega$ is connected we get $\displaystyle u$ constant a.e. in $\displaystyle \Omega$.

    Assume the result for $\displaystyle m\leq k$, we want to prove it for $\displaystyle k+1$:

    Assume for a moment that we know that if $\displaystyle f: \Omega \rightarrow \mathbb{R}$ satisfies the hypothesis with the additional $\displaystyle f\in C^{\infty } (\Omega )$ then $\displaystyle f\in \mathbb{R} [x_1,...,x_n]$ with $\displaystyle \deg (f) \leq k-1$. Now let $\displaystyle u \in W_{loc}^{k+1,p} (\Omega )$ as in the hypothesis and $\displaystyle V \subset \subset \Omega$ then it's clear that there exists a subsequence $\displaystyle \varepsilon _l \rightarrow 0$ such that $\displaystyle D^{\beta}u_{\varepsilon _l}=\mbox{constant} = D^{\beta }u$ for all $\displaystyle |\beta |= k$ (by the case $\displaystyle k=1$), but then $\displaystyle u-u_{\varepsilon _l} \in W^{k,p}(V)$ and $\displaystyle D^{\beta }(u-u_{\varepsilon _l})=0$ for all $\displaystyle |\beta |=k$ and so $\displaystyle u-u_{\varepsilon _l} \in \mathbb{R} [x_1,...,x_n]$ and $\displaystyle \deg (u-u_{\varepsilon _l}) \leq k-1$, but since $\displaystyle D^{\alpha } (u_{\varepsilon } )= (D^{\alpha }u)_{\varepsilon }$ we get that $\displaystyle u_{\varepsilon _l} \in \mathbb{R} [x_1,...,x_n]$ and $\displaystyle \deg (u_{\varepsilon _l}) \leq k$. This gives that $\displaystyle u\in \mathbb{R} [x_1,...,x_n]$ and $\displaystyle \deg (u) \leq k$. This proves that $\displaystyle u$ is a polynomial in $\displaystyle V$, but in particular it is analytic and $\displaystyle \Omega$ is connected, so if we cover $\displaystyle \Omega$ with compactly contained sets (countably many), we get that each polynomial representation of $\displaystyle u$ is identical in every subset (since polynomials are analytic, intersection of (finitely) open sets is open and the fact that $\displaystyle \Omega$ is connected).

    Now to prove the first part of the last paragraph it is sufficient to look at $\displaystyle V\subset \subset \Omega$ where $\displaystyle V=\{ (a_1,b_1) \times ...\times (a_n,b_n) : a_i,b_i \in \mathbb{R}, \ a_i<b_i \}$. Let $\displaystyle x=(x_1,x_2)\in \mathbb{R} ^{n-1} \times \mathbb{R}$ and take $\displaystyle f$ as assumed, for fixed $\displaystyle x_1$ let $\displaystyle f_{x_1}(x_2) = f(x_1,x_2)$ where $\displaystyle f_{x_1} : (a_n,b_n) \rightarrow \mathbb{R}$ (at his point it's obvious we're using an inductive argument on the dimension and the case $\displaystyle n=1$ is trivial). Since $\displaystyle f_{x_1}^{(k)} \equiv 0$ we get that $\displaystyle f_{x_1} (x_2)= \sum_{j=1}^{k-1} c_j(x_1)x_2^{j}$. The smoothness of $\displaystyle f$ implies the smoothness of the coefficients $\displaystyle c_j(x_1)$. Now if $\displaystyle |\beta | =k$ with $\displaystyle \beta \in \mathbb{N} ^{n-1}$ then $\displaystyle \sum_{j=1}^{k-1}x_2^jD^{\beta }c_j(x_1)= D^{\beta } f =0$, so by induction $\displaystyle c_j(x_1)$ are polynomials in $\displaystyle n-1$ variables with $\displaystyle \deg (c_j) \leq k-1$. This gives us that $\displaystyle f$ is a polynomial in $\displaystyle n$ variables with $\displaystyle \deg (f) \leq 2(k-1)$, but the condition $\displaystyle D^{\alpha } f=0$ for all $\displaystyle |\alpha | =k$ gives that $\displaystyle \deg (f) \leq k-1$.

    Any comments and suggestions are appreciated.

    Edit: It is worth noting that in the thoery of Sobolev spaces $\displaystyle W^{k,p}(\Omega)$ there are some useful embeddings for some $\displaystyle \Omega$ (like, for example, if $\displaystyle n>p$ we have $\displaystyle W^{k,p}(\mathbb{R} ^n) \hookrightarrow C^{0,\gamma }(\mathbb{R} ^n)$ this last one being the space of Hölder cont. functions with exponent $\displaystyle \gamma =1-\frac{n}{p}$) with certain properties, and one corollary of these embeddings is that if $\displaystyle u\in W^{k,p}(\Omega )$ for all $\displaystyle k\in \mathbb{N}$ then $\displaystyle u\in C^{\infty }(\Omega )$ which makes most of this proof unnecessary.
    Last edited by Jose27; Apr 25th 2010 at 06:11 PM.
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