All the following we use Evans notation.

By Green's reconstruction formula, we could represent $u$ by $$ u(x)=\int_\Omega-\triangle u(y)G(x,y)dy-\int_{\partial \Omega}u(y)\partial_\nu G(x,y)d\sigma(y) $$ where $\Omega$ is open bounded with smooth boundary and $\nu$ is the outer normal vector, $G(x,y)$ is the Green function.

The book gives the prove of this formula based on the fact that $u\in C^2(\bar{\Omega})$. My professor says that by standard approximation we could have the result for $C^2(\Omega)\cap C^0(\bar{ \Omega})$

I got stuck on how this approximation works. What kind of approximation should I use here?

And my second question: Suppose now $u=0$ on $\partial\Omega$, then the formula $$ u(x)=\int_\Omega-\triangle u(y)G(x,y)dy$$ should hold such that $u(x)\to 0$ as $x\to\partial\Omega$. However, I can't prove it...

Please give me some hints!

Thx!

All the following we use Evans notation.

By Green's reconstruction formula, we could represent $u$ by $$ u(x)=\int_\Omega-\triangle u(y)G(x,y)dy-\int_{\partial \Omega}u(y)\partial_\nu G(x,y)d\sigma(y) $$ where $\Omega$ is open bounded with smooth boundary and $\nu$ is the outer normal vector, $G(x,y)$ is the Green function.

The book gives the prove of this formula based on the fact that $u\in C^2(\bar{\Omega})$. My professor says that by standard approximation we could have the result for $C^2(\Omega)\cap C^0(\bar{ \Omega})$

I got stuck on how this approximation works. What kind of approximation should I use here?

And my second question: Suppose now $u=0$ on $\partial\Omega$, then the formula $$ u(x)=\int_\Omega-\triangle u(y)G(x,y)dy$$ should hold such that $u(x)\to 0$ as $x\to\partial\Omega$. However, I can't prove it...

Thank you.