In page $36$ of "Partial Differential Equation," Evan define $v(z) = \Phi(z-x) - \phi^x(z)$, where $\phi^x(z)$ is a corrector function, satisfying the following identities,

$$ \Delta \phi^x = 0 \ in \ U \tag{1}$$ $$ \phi^x = \Phi(y-x) \ in \ \partial U \tag{2}$$

$$\lim_{\epsilon \rightarrow 0} \int_{\partial B(x,\epsilon)} \frac{\partial v}{\partial \nu} w dS = \lim_{\epsilon \rightarrow 0} \int_{\partial B(x,\epsilon)} \frac{\partial \Phi}{\partial \nu}(x-z) w(z) dS $$

I am kind of lost about how he got the equation.


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