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I am attempting to solve some problems from Evans , I need some help with the following question. If $u\in H^2_0\Omega$, where $\Omega$ is open, bounded subset of $R^n$. How can i solve biharmonic equation

$\triangle^2u=f$ in $\Omega$, $u =\frac {\partial u } {\partial n }=0$ on $\partial \Omega$, $n$ is the normal vector .

provided $\int _\Omega \triangle u \triangle v dx =\int _\Omega fv $ for all $v\in H^2_0\Omega$. Given $f \in L^2(\Omega)$ , and prove that the weak solution is unique . Any kind of help would be great .

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$u=\frac{\partial u}{\partial n} = 0$ on $\partial \Omega$? Instead of $u \frac{\partial u}{\partial n}$? –  user20266 Jun 10 '12 at 7:44
    
@Thomas : Thank you for pointing out the mistake . –  Theorem Jun 10 '12 at 7:58
    
Do you still need an answer for this question? –  Shuhao Cao Jun 18 '13 at 21:35
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4 Answers

Four hints:

i) what kind of functional is $v\mapsto \int fv$? It's obviously linear, but is it continuous?

ii) Assume $u,\bar{u}$ solve the problem. Then $\int\Delta(u-\bar{u})\Delta v dx =0$ for all admissible $v$. What is the image of $\Delta$ when applied to admissible $v$? I.e. for which $\phi$ can you solve $\Delta v = \phi$? All these $\phi$ are admissible test functions. What does that tell you about $u-\bar{u}$?

iii) What does ii) tell you about $(u,v)\mapsto \langle u, v\rangle := \int\Delta u \Delta v dx$ ? Could this possibly be a scalar product? If yes, on which space?

iv) Now try to combine i) and iii). Does any representation theorem for linear functionals apply?

This comes without any kind of warranty, I do explicitly state that I did not check the details, it's just the roadmap I'd try first (with quite some confidence, I'd like to add, though). But you asked for assistance not for a solution :-)

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Thank you very much :) . First I will try with the hints. –  Theorem Jun 10 '12 at 8:53
    
I have been trying to get some idea but i am not able at all . If you could help me with detailed solution it would be great . –  Theorem Jun 10 '12 at 10:20
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The Lax-Milgram theorem

Given a Hilbert space $V$ with scalar product $(.,.)_V$ and corresponding norm $\|\cdot\|_V$, a continuous and coercive bilinear form $a(.,.)$ on $V \times V$ and a continuous linear functional $L$ on $V$, there exists unique $u \in V$ s.t.

$a(u,v) = L(v)\ \ \ \ \forall v \in V$.

Some of the proofs required if you want to use this:

  • $a(.,.)$ is symmetric, ie, it holds that $a(u,v) = a(v,u) \ \ \ \forall u,v \in V$
  • $a(.,.)$ is continuous, ie, there exists a constant $C>0$ s.t. $|a(u,v)|\leq C\|u\|_V\|v\|_V \ \ \forall u,v \in V $
  • $a(.,.)$ is coercive or V-elliptic, ie, there exists a constant $\alpha>0$ s.t. $a(u,u) \geq \alpha \|u\|^2_V \ \ \ \forall u \in V$
  • $L$ is continuous, ie, there exists a constant $\Lambda>0$ s.t. $|L(v)| \leq \lambda \|v\|_V \ \ \ \forall v \in V$

I hope this gets you started.

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I have some ideas about the first question.You can use the method that we find the solution of the Possion's Equation.

First, you can find a spherical symmetry solution of the biharmonic equation. Thus, you will get a fundamental solution. Then you can get solution of $\Delta^2u=f$ by Green's Identity.

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For the second question,you can use the representation theorem in Hilbert Space.For more details,you can read Page 118 in the PDE by Fritz John. –  Juntao Huang Jun 10 '12 at 8:34
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We will use the Lax-Milgram Theorem. A weak solution of your problem is a $u\in H^2(\Omega)$ such that

\begin{eqnarray} \Delta^2u=f & \Rightarrow & [\Delta^2u]\varphi=f\varphi\\ & \Rightarrow & \int_{\Omega}\Delta(\Delta u)\varphi=\int_{\Omega} f\varphi\\ & \Rightarrow & -\int_{\Omega}\nabla(\Delta u)\nabla\varphi=\int_{\Omega} f\varphi\\ & \Rightarrow & \int_{\Omega}\nabla u\nabla\varphi=\int_{\Omega} f\varphi, \end{eqnarray} for all $\varphi\in H^2_0(\Omega)$. Define the bilinear operator $B:H^2_0(\Omega)\times H^2_0(\Omega)\rightarrow\mathbb{R}$, $$B(u,\varphi)=\int_{\Omega}\Delta u\Delta\varphi.$$ Statement 1 This bilinear operator is continuos.

In fact,

\begin{eqnarray} |B(u,\varphi)| & \leq & \int_{\Omega}|\Delta u||\Delta\varphi|\\ & \leq & \|\Delta u\|^2_{L^2(\Omega)}\|\Delta \varphi\|^2_{L^2(\Omega)}\\ & \leq & C\|u\|^2_{H^2_0(\Omega)}\|\varphi\|^2_{H^2_0(\Omega)} \end{eqnarray} You can prove easily this last inequality.

Statemant 2 The bilinear operator is coercive.

In fact, $$B(u,u)=\int|\Delta u|^2=\|\Delta u\|^2_{L^2(\Omega)}\geq C\|u\|^2_{H^2_0(\Omega)}.$$ We used that $\|\Delta u\|_{L^2(\Omega)}$ defines a norm on $H^2_0(\Omega)$ equivalent to the usual norm.

Then, by the Lax-Milgram Theorem, for each $f\in H^2_0(\Omega)$, exists an unique function $u\in H^2_0(\Omega)$ such that $$B(u,\varphi)=\int_{\Omega}\Delta u\Delta\varphi=\int_{\Omega}f\varphi,$$ for all $\varphi\in H^2_0(\Omega)$.

I hope I help you.

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Proof of $\|\Delta u\|_{L^2} \geq \|u\|_{H^2}$ is totally non-trivial! You might wanna address that problem. For you used $\|\Delta u\|_{L^2} \geq \|D^2 u\|_{L^2}$, $\|\Delta u\|_{L^2} \geq \|\nabla u\|_{L^2} \leq \|u\|_{L^2}$. And a suggestion on notation is: there is no such term as "$H^2_0(\Omega)$-norm" in your subscript for norms, only $H^2$-norm. –  Shuhao Cao Jun 16 '13 at 7:40
    
Thank you for this observations. You are right. I knew that the proof of this inequality is non-trivial, but as I dont know to prove this, I wrote just what I did in my exercise. –  José Carlos Jun 17 '13 at 4:12
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