### Archive

Posts Tagged ‘functional analysis’

## Agregation 2013 – Analysis – Part 4

Part IV: To be or not to be with separable dual

1. Consider ${S=[a,b]}$ a segment (${a) of ${[0,1]}$ and ${V}$ a closed linear subspace of ${C([0,1])}$ such that each function in ${V}$ is of class ${C^1}$ on ${S}$. For ${(x,y) \in S^2}$ such that ${x \neq y}$ we denote ${\displaystyle \xi_{x,y}(f)=\frac{f(x)-f(y)}{x-y}}$ where ${f \in V}$.

(a) Prove that ${\xi_{x,y} \in V^*}$.

(b) Prove that for every ${f \in V}$ we have

$\displaystyle \sup_{\substack{x,y \in S\\ x\neq y}} |\xi_{x,y}(f)|<\infty.$

(c) Deduce that there exists ${\mathcal{N}(S)>0}$ which verifies for each ${f \in V}$ and ${(x,y) \in S^2}$

$\displaystyle |f(x)-f(y)| \leq \mathcal{N}(S) |x-y|\|f\|_\infty.$

(d) Let ${(t_l)_{0\leq l \leq L}}$ be a finite sequence of points in ${S}$ such that ${0 (for ${0\leq l ) and ${t_0=a,t_L=b}$. Prove that

$\displaystyle \forall f \in V,\ \sup_{t \in S}|f(t)| \leq \sup_{0\leq l\leq L}|f(t_l)|+\frac{1}{2}\|f\|_\infty$

2. Let ${F_0}$ be a closed linear subspace of ${C([0,1])}$ such that every function in ${F_0}$ is of class ${C^1}$ on ${[0,1]}$. Prove that ${F_0}$ has finite dimension.

## 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

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

## Weak formulation for Laplace Equation with Robin boundary conditions

Consider ${\Omega \subset \Bbb{R}^N}$ an open set with Lipschitz boundary and consider on ${\Omega}$ the following problem

$\displaystyle \begin{cases} -\Delta u =f &\text{ in }\Omega \\ \frac{\partial u}{\partial n}+\beta u =0 & \text{ on }\partial \Omega. \end{cases}$

where ${\beta>0}$ is a constant. This is the Laplace equation with Robin boundary conditions. I will prove that the problem is well posed and for each ${f \in L^2(\Omega)}$ there exists a solution ${u \in H^2(\Omega)}$.

## Lax Milgram application

Let ${I=(0,2)}$ and ${V=H^1(I)}$. Consider the bilinear form

$\displaystyle a(u,v)= \int_0^2 u'(t)v'(t)dt +\left( \int_0^1 u(t)dt\right)\left( \int_0^1 v(t)dt\right).$

1. Check that ${a(u,v)}$ is a continuous symmetric bilinear form and that ${a(u,u)=0}$ implies ${u=0}$.

2. Prove that ${a}$ is coercive.

3. Deduce that for every ${f \in L^2(I)}$ there exists a unique ${u \in H^1(I)}$ satisfying

$\displaystyle (1) \ \ \ \ a(u,v)=\int_0^2 fv,\ \forall v \in H^1(I).$

What is the corresponding minimization problem?

4. Show that the solution of ${(1)}$ belongs to ${H^2(I)}$ (and in particular ${u \in C^1(\overline{I})}$). Determine the equation and the boundary conditions satisfied by ${u}$.

5. Assume that ${f \in C(\overline{I})}$, and let ${u}$ be the solution of ${(1)}$. Prove that ${u}$ belongs to ${W^{2,p}(I)}$ for every ${p<\infty}$. Show that ${u \in C^2(\overline{I})}$ if and only if ${\int_If=0}$.

6. Determine explicitly the solution ${u}$ of ${(1)}$ when ${f}$ is a constant.

7. Set ${u=Tf}$, where ${u}$ is the solution of ${(1)}$ and ${f \in L^2(I)}$. Check that ${T}$ is a self-adjoint compact operator from ${L^2(I)}$ into itself.

8. Study the eigenvalues of ${T}$.

H. Brezis, Functional Analysis