Question regarding Laplace equation under Evans setting

Question

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.

Answer 1

I think usually the standard approximation arguments gives you the result $C^2(\Omega)\cap C^1(\bar{\Omega})$ and I did not quit see how to reduce to $C^2(\Omega)\cap C^0(\bar{\Omega})$.

However, your second question is a good exercise for you to understand the properties of the Green's function. Roughly speaking, $\Delta G(x,y)=\delta_{x-y}$, by the properties of Dirac measure, you can solve the linear equation via (zero boundary case) $$u(x)=\int_\Omega u(y)\Delta G(x,y)dy=\int_\Omega \Delta u(y)G(x,y)dy.$$ When $u=0$ on $\partial\Omega$, the above equation clearly tells you $u(x)\to0 $ as $x\in\partial\Omega$.

All these facts I mentioned are rather standard and can be found for instance in nice book of

Han, Qing; Lin, Fanghua Elliptic partial differential equations. Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997. x+144 pp.