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I have a problem:

For $\Omega$ be a bounded domain in $\Bbb R^n$. We consider $$\left\{\begin{matrix} \Delta u-\lambda u =f \ \rm in \ \Omega & \\ u\mid_{\partial {\Omega}} =0 \end{matrix}\right. \tag I$$

  • a/ Definition of classical solution;

  • b/ Definition of weak solution in the $W^{1,2}(\Omega)$ space.

Finding the conditions so that weak solution becomes classical solution.

  • c/ Prove that if $\lambda=0$ then $(I)$ has a unique solution in $W^{1,2}(\Omega)$;

  • d/ Determine the existence and uniqueness of solution of $(I)$ by $\lambda$.

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I'm not sure but I still write it here:

*) For a): Definition of classical solution

Let $f \in C^{0}(\Omega)$. A function $u \in C^{2}(\Omega) \cap C^0(\overline{\Omega})$ such that $(I)$ forall $x \in \overline{\Omega}$.

*) For b): Definition of weak solution in the $W^{1,2}(\Omega)$ space

Let $f \in H^{-1}(\Omega)=(H_{0}^{1}(\Omega))^*$. A function $u$ such that:

$$\Delta u - \lambda u=f \ \rm in \ \Omega \iff \left \langle \Delta u - \lambda u,\varphi \right \rangle=\left \langle f,\varphi \right \rangle \ \text{for any test function} \ \varphi \in H_{0}^{1}(\Omega)$$

  • Finding the conditions so that weak solution becomes classical solution:

For $f \in H^k(\Omega)$ and $\partial \Omega \subset C^{k+2}(\Omega)\implies u \in H^{k+2}(\Omega)\cap H_{0}^{1}(\Omega)$.

If $k+2>\dfrac{n}{2}+2 \implies k>\dfrac{n}{2}$. By applying Sobolev embedding theorem, we have $u \in C^2(\overline{\Omega})$. Because $$H^{k+2}(\Omega)\overset{\rm embedding}{\rightarrow} C^2(\overline{\Omega})$$...

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Now, I have stuch when I go on...Can anyone help me!

Any help will be appreciated! Thanks!

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What techniques do you know to prove existence of solutions for this type of PDE? –  Tomás Nov 7 '13 at 11:01
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To prove existence, you have know some technique. There are plenty of them: Minimization, Fredholm Alternative, Monotone Operators and so on. Do you know any of these approachs? –  Tomás Nov 7 '13 at 13:03
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To help you, I need to know your background. Are you making some course? Are you studying alone? –  Tomás Nov 7 '13 at 13:13
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What are you studying in this course? From previous questions it seems that you are studying Fredholm alternative. Is this true? –  Tomás Nov 7 '13 at 13:19
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Yes they are right. Just one thing, there is no need to say that $f\in C^0(\Omega)$. Just say that $u$ is a classical solution if it belongs to $C^0(\Omega)\cap C^2(\overline{\Omega})$ and satisfies $(I)$. –  Tomás Nov 7 '13 at 13:28

1 Answer 1

up vote 2 down vote accepted

You have said in the comments that c) is solved, hence I will start from it. Let $S:L^2(\Omega)\to H_0^1(\Omega)$ be the solution operator associated with the problem $$ \left\{ \begin{array}{ccc} -\Delta u=f &\mbox{ in $\Omega$} \\ u=0 &\mbox{ in $\partial\Omega$} \end{array} \right.\tag{1} $$

i.e. if $Sf=u$ then, $u$ solves $(1)$ in the weak sense. Note that $-\Delta (Sf)=f$ for $f\in L^2(\Omega)$. We apply Rellich–Kondrachov theorem to conclude that $S:L^2(\Omega)\to L^2(\Omega)$ is compact. I wil leave to you to prove that $S$ is symmetric, which implies that $S=S^\star$.

Consider the problem $$f+\lambda Sf=u\tag{2}$$

where $f,u\in L^2(\Omega)$. Assume that for $u=0$ the only solution of $(2)$ is $f=0$, then the Fredholm alternative implies that for each $u\in L^2(\Omega)$, problem $(2)$ has a unique solution. Take $u\in H_0^1(\Omega)$ and note that equation $(2)$ implies that $f\in H_0^1(\Omega)$. Apply $-\Delta$ in both sides to get $$-\Delta f+\lambda f=-\Delta u\tag{3}$$

Because the range of $-\Delta $ is $L^2(\Omega)$, we conclude from $(3)$ that for each $g\in L^2$ there exist unique $u\in H_0^1(\Omega)$ such that $$-\Delta u+\lambda u=g$$

Now we pass to the second alternative, i.e. assume that for $u=0$, there is a finite dimensional subspace of $L^2(\Omega)$ with solutions of $(2)$. Apply the Fredholm alternative. What can you conclude now?

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Yes, you're very good at maths! Thanks Tomás! –  kimtahe6 Nov 9 '13 at 16:56
    
I think for question b/ Definition of weak solution in the $W^{1,\ 2}(\Omega)$ space. We need $ u \in W_0^{1,\ 2}(\Omega)$ and $\int_{\Omega}\nabla u \overline{\nabla \eta}\rm dx + \lambda \int_{\Omega}u \overline{\eta} \rm d x=-\int_{\Omega}f \overline{\eta}\rm dx, \forall \eta \in W_0^{1,\ 2}(\Omega)$????? –  kimtahe6 Dec 11 '13 at 16:22
    
Yes this is the definition. Are you considering complex functions or real functions? –  Tomás Dec 11 '13 at 18:35
    
Finding the conditions so that weak solution becomes classical solution: I rewrite as :) $$\int_{\Omega}\nabla u \overline{\nabla \eta}\rm dx + \lambda \int_{\Omega}u \overline{\eta} \rm d x=-\int_{\Omega}f \overline{\eta}\rm dx, \forall \eta \in W_0^{1,\ 2}(\Omega)$$ Whence $$\left \langle \Delta u, \eta \right \rangle-\lambda\left \langle u, \eta \right \rangle=\left \langle f,\eta \right \rangle$$ So $\Delta u-\lambda u =f $, because $u$ is continuous. Moreover, $u \mid_{\partial \Omega}=0$ (with $u \in C(\overline {\Omega})$) –  kimtahe6 Dec 12 '13 at 14:11
    
For question c): We assume that $u_1, u_2$ are solutions of (I) when $\lambda=0$. Let $u:=u_1-u_2 \in W_{0}^{1,2} \implies \Delta u =0$. We have $$\left ( \Delta u,u \right )=\sum \int \dfrac{\partial u}{\partial x_i} \overline{\dfrac{\partial u}{\partial x_i}}dx=\left \| \nabla \right \|^2_{L^2} \ge C_{\Omega}^{-2}\left \| u \right \|^2_{L^2}$$ (Friedrichs'ineq). So $\left \| u \right \|_{W^{1,2}}=0 \implies u=u_1-u_2=0.$ Is it correct, Tomás? –  kimtahe6 Dec 12 '13 at 14:19

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