### Archive

Posts Tagged ‘sobolev’

## Characterizations of Sobolev Spaces

October 26, 2012 1 comment

I will present here a few useful characterizations of Sobolev spaces. The source is Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis. Consider ${u \in L^p(\Omega)}$ with ${1 < p \leq \infty}$. We denote by ${p'}$ the conjugate of ${p}$, i.e. ${\displaystyle \frac{1}{p}+\frac{1}{p'}=1}$. Then the following properties are equivalent:

• (i) ${u \in W^{1,p}(\Omega)}$;
• (ii) there exists a constant ${C}$ such that

$\displaystyle \left| \int_\Omega u \frac{\partial \varphi}{\partial x_i} \right| \leq C \| \varphi\|_{p'} ,\ \forall \varphi \in C_c^\infty(\Omega),\ \forall i=1..N;$;

• (iii) there exists a constant ${C}$ such that for all ${\omega \subset \subset \Omega}$ and all ${h \in \Bbb{R}^N}$ with ${|h|<\text{dist}(\omega,\partial \Omega)}$ we have

$\displaystyle \| \tau_hu-u\|_{L^p(\omega)}\leq C|h|.$

(note that ${\tau_hu(x)=u(x+h)}$ makes sense for ${x \in \omega}$ and ${|h|<\text{dist}(\omega,\partial \Omega)}$. Furthermore, we can take ${C=\|\nabla u\|_{L^p(\Omega)}}$ in (ii) and (iii). If ${\Omega=\Bbb{R}^N}$ we have

$\displaystyle \|\tau_hu-u\|_{L^p(\Bbb{R}^N)} \leq |h|\|\nabla u\|_{L^p(\Bbb{R}^N)}$.

## One dimensional trace of a Sobolev function

Check that the mapping ${u\mapsto u(0)}$ from ${H^1(\Bbb{R})}$ to ${\Bbb{R}}$ is a continuous linear functional on ${H^1(0,1)}$. Deduce that there exists a unique ${v_0 \in H^1(0,1)}$ such that

$\displaystyle u(0)=\int_0^1(u'v_0'+uv_0).$

Show that ${v_0}$ is the solution of some differential equation with appropriate boundary conditions and compute ${v_0}$ explicitly.

H. Brezis, Functional Analysis, Ex 8.18

## A lemma of J. L. Lions

Let $X,Y$ and $Z$ be three Banach spaces with norms $\|\cdot \|_X,\ \|\cdot \|_Y$ and $\|\cdot \|_Z$. Assume that $X \subset Y$ with compact injection and that $Y\subset Z$ with continuous injection. Prove that

$\forall \varepsilon >0 \exists C_\varepsilon \geq 0$ satisfying $\|u\|_Y \leq \varepsilon \|u\|_X+C_\varepsilon \|u\|_Z,\ \forall u \in X$.

Applications:

1. Prove that for every $\varepsilon >0$ there exists $C_\varepsilon \geq 0$ satisfying$\displaystyle \max_{[0,1]}|u| \leq \varepsilon \max_{[0,1]}|u^\prime|+C_\varepsilon\|u\|L^1,\ \forall u \in C^1([0,1])$.
2. Pick $p>1$. Prove that for every $\varepsilon >0$ there exists $C=C(\varepsilon,p)$ such that $\|u\|_{L^\infty(0,1)} \leq \varepsilon \|u^\prime\|_{L^p(0,1)}+C\|u\|_{L^1(0,1)},\ \forall u \in W^{1,p}(0,1)$.

Source: Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011

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## Sobolev space impossible extension

Let $\Omega=\{ (x,y) \in \Bbb{R}^2 : x \in (0,1),\ y \in (0,x^2)\}$. And for $\beta < 3/2$ (!!! Correction: $1<\beta<3/2$) consider $v: \Omega \to \Bbb{R},\ v(x,y)=x^{1-\beta}$. Prove that:

1) $\Omega$ does not have Lipschitz boundary (i.e. its boundary is not locally the graph of Lipschitz functions).

2) $v \in H^1(\Omega)$.

3) For every ball $B$ which contains $\Omega$, there is no function in $H^1(B)$ which extends $v$.

This problem gives a counter example which states that if $\Omega$ doesn’t have Lipschitz boundary, then there may be no extension to some functions in $H^1(\Omega)$ to greater Sobolev spaces.

## Generalized Poincare Inequality

Consider $\Omega \subset \Bbb{R}^N$ a bounded domain with Lipschitz boundary. If $H$ is a non-zero, closed subspace in $H^1(\Omega)$, which does not contain the non-zero constant functions, then there is a constant $C>0$, depending on $\Omega$, such that $\|u \|_{L^2}\leq C \| |\nabla u | \|_{L^2},\ \forall u \in H$.
Note that this generalizes the usual Poincare inequality, which says that the above inequality holds for some $C>0$ on the space $H_0^1(\Omega)$, a space which does not contain the non-zero constant functions.
Let $u \in H^s(\Bbb{R}^n)$ with $s > n/2$. Show that $\lim_{|x| \to \infty} u(x)=0$,