Definition of weak solution in $W^{1,2}(\Omega)$.

Question

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!

Answer 1

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?