Establishing a relationship between weak solution in $L_2(\Omega)$ and weak solution in $W^{1,\ 2}(\Omega)$ with classical solution

I have a problem:

For $\Omega$ be a bounded domain in $\Bbb R^n$.

Denote $L_{2,\ 0}(\Omega)=\overline{C_{0}^{\infty}(\Omega)}$, with norm in $L_2(\Omega)$.

We consider: $$\left\{\begin{matrix} \Delta u +u_x+au=f(x)\ \text{in}\ \Omega& \\ \ \ \ \ \ \ \ \ \ \ \ u\mid_{\partial \Omega}=0\ \text{on}\ \Omega & \end{matrix}\right. \tag I$$

a/ Definition of weak solution of (I) in $L_2(\Omega)$;

b/ Establishing a relationship between weak solution in $L_2(\Omega)$ and weak solution in $W^{1,\ 2}(\Omega)$ with classical solution.

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We have:

$\Delta u +u_x+au=f \iff \int_{\Omega}\Delta u \cdot v \rm dx +\int_{\Omega} u_x v \rm dx +\int_{\Omega}au \cdot v \rm dx =\int_{\Omega}fv \rm dx$

On the other hand, \begin{align}\int_{\Omega}\Delta u \cdot v \rm dx &=\sum_{i=1}^{n}\int_{\Omega}\dfrac{\partial^2 u}{\partial x_i^2}vdx\\ &=\int_{\partial \Omega}\sum_{i=1}^{n}\dfrac{\partial u}{\partial x_i}v\cos(n,x_i)dx-\int_{\Omega}\sum_{i=1}^{n}\dfrac{\partial u }{\partial x_i}\dfrac{\partial v}{\partial x_i}dx\\ &=-\int_{\Omega}\sum_{i=1}^{n}\dfrac{\partial u }{\partial x_i}\dfrac{\partial v}{\partial x_i}dx \end{align} Whence,

$$-\int_{\Omega}\sum_{i=1}^{n}\dfrac{\partial u }{\partial x_i}\dfrac{\partial v}{\partial x_i}dx+\int_{\Omega} u_x v \rm dx +\int_{\Omega}au \cdot v \rm dx =\int_{\Omega}fv \rm dx$$

It means that $$\boxed{\int_{\Omega}\nabla u\cdot \nabla v \rm dx-\int_{\Omega}u_xvdx-\int_{\Omega}au \cdot vdx=-\int_{\Omega}fvdx}$$

Now, I have stuch although I have tried...

Any help will be appreciated! Thanks!

• Are you sure the norm is correct? With this norm $L_{2,0}(\Omega)=L^2(\Omega)$ where $L^2(\Omega)$ is the usual Lebesgue space. – Tomás Dec 11 '13 at 18:40
• I haven't posted in a long time. But I still remember that you are very good at math :). How r u? Tomás! – kimtahe6 Dec 12 '13 at 5:13
• @ Tomás: $L_{2,\ 0}(\Omega)=\overline{C_{0}^{\infty}(\Omega)}$ with norm of $L_2(\Omega)$ (we understand here $L_2$ is $L^2$). I think that it does not affect this problem. Because the question is: "Establishing a relationship between weak solution in $L_2(Ω)$ and weak solution in $W^{1, 2}(Ω)$ with classical solution."? And can you help me? – kimtahe6 Dec 12 '13 at 5:25
• What is a weak solution in $L_2(\Omega)$? There is no sense in talk about such a thing. You need at least one derivative to use integration by parts. – Tomás Dec 12 '13 at 11:18
• I'm sorry but it's in my book :( – kimtahe6 Dec 12 '13 at 13:57

a) Let the notation $u_x$ imply some partial derivative, say, $\frac{\partial u}{\partial x_m}.$ The following three definitions are standard in the $L^p$-theory of boundary value problems for PDE.

Definition 1. For a bvp (I), a strong solution of the class $W^{2,2}$ is an element $u\in W^{2,2}(\Omega)$ satisfying the equation (I) a.e. in $\Omega$, with the boundary condition (I) understood in terms of traces.

Definition 2. For a bvp (I), a weak solution of the class $L^2$ is an element $u\in L^2(\Omega)$ satisfying the integral identity $$\int\limits_{\Omega}u\Delta\,v\,dx-\int\limits_{\Omega}u\frac{\partial v}{\partial x_m}\,dx+\int\limits_{\Omega}auv\,dx=\int\limits_{\Omega}fv\,dx\quad\forall\, v\in W^{2,2}(\Omega)\,\colon \;\,v|_{\partial\Omega} =0\tag{\ast}$$ Remark. If the boundary condition were inhomogeneous, say, $u|_{\partial\Omega}=\varphi$, the left-hand side of the identity $\,(\ast)\,$ should contain an additional term $$-\int\limits_{\partial\Omega}\varphi\frac{\partial v}{\partial n}\,ds$$ with an outward unit normal $n$ to $\partial\Omega$.

Definition 3. For a bvp (I), a definition of a weak solution of the class $L^2$ is rated as correct iff
i) a strong solution of the class $W^{2,2}$ will be a weak solution of the class $L^2$;
ii) a weak solution of the class $L^2$ possessing the smoothness of $W^{2,2}$ will be a strong solution.

Thus for Defnition 2, being correct just implies that the integral identity $\,(\ast)\,$ does contain the required equation and boundary condition. Checking the correctness of Definition 2 reduces to a not very difficult exercise. Besides the fact that $C_0^{\infty}(\Omega)$ is dense in $L^2(\Omega)$, you need to know that for a domain $\Omega$ with a sufficiently smooth boundary, holds the following extension theorem $$\forall\,\varphi_0\in W^{s-\frac{1}{2},2}(\partial\Omega),\;\forall\,\varphi_1\in W^{s-\frac{3}{2},2}(\partial\Omega)\;\exists\; \psi\in W^{s,2}(\Omega)\;\colon\; \psi|_{\partial\Omega}=\varphi_0,\;\frac{\partial\psi}{\partial n}\biggr|_{\partial\Omega}=\varphi_1\,,$$ where $s>\frac{3}{2}\,$,  though $\,s=2\,$ is quite enough here. You will need this this extension theorem to extract the boundary condition $\,u|_{\partial\Omega}=0\,$ from the integral identity $\,(\ast)\,$.

b) Establishing a relationship between weak solutions of the class $L^2$ and classical solutions is a rather technical matter based on the so-called localization techniques that cannot fit this format.