I have the following identity

If $\Omega$ is a smooth bounded domain and $u\in C^2(\bar\Omega)$ then for $x\in \Omega$ $$u(x)=\int\limits_\Omega \Phi(x-y)\Delta u(y)dy+\int\limits_{\partial\Omega}\big(u(y)\frac{\partial\Phi}{\partial n}(x-y)-\Phi(x-y)\frac{\partial u}{\partial n}(y)\big)dS_y$$

where $\Phi$ is the fundamental solution to the Laplace equation $\Delta u=0$.

When I tried to check the legitimacy of the formula I faced a problem :

For $u=1$ I get $$1=\int\limits_{\partial\Omega}\frac{\partial\Phi}{\partial n}(x-y)dS_y=\int\limits_{\Omega}\Delta\Phi(x-y)dy=0$$ which I got by (taking $v=1$ in) Green's Identity : $$\int\limits_{\Omega}\Delta u(x)dx=\int\limits_{\partial\Omega}\frac{\partial u}{\partial n}(x)dS_x$$ What went wrong?


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