Posts Tagged ‘functional analysis’

Agregation 2013 – Analysis – Part 4

June 24, 2013 Leave a comment

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

1. Consider {S=[a,b]} a segment ({a<b}) 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<t_{i+1}-t_i\leq \displaystyle\frac{1}{\mathcal{N}(S)}} (for {0\leq l <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.

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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)}.

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Weak formulation for Laplace Equation with Robin boundary conditions

October 22, 2012 10 comments

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)}.

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Lax Milgram application

October 14, 2012 Leave a comment

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

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Uniqueness and Error estimates via Kinetic Entropy Defect Measure

March 7, 2012 Leave a comment

Here are a few thoughts from my preparation for the exam of Kinetic Equations at Universite de Savoie, France. The teachers of the course were Christian Bourdarias and Stephane Gerbi. I had to study an article of Benoit Perthame entitled Uniqueness and Error estimates in First Order Quasilinear Conservation Laws via the Kinetic Entropy Defect Measure.

This was a very nice article to study, since it used many things like distribution theory, measures and regularization. It showed the power of these tools, and motivated me to learn more about them.

As the title of the article says, a relatively new proof of the uniqueness of the solution for a scalar conservation law coupled with some entropy inequalities is given. The only known proof at the time the article was published was due to Kruzkov and was more intricate and difficult to understand than the one provided in the article. The estimates on the entropy defect measure, which will be introduced can yield some error term approximation for approximate equation, which in particular imply unicity at once.

Here are my detailed notes on the article. They are handwritten, but I think they are readable. Perthame-Uniqueness and Error Estimates

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