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Let $B_+$ and $B$ denote the upper half and full unit disc in $\mathbb{R}^2$ respectively. Suppose $f \in L^2(B_+)$ and that $u \in H_0^1(B_+)$ is a weak solution to the Dirichlet BVP $-\Delta u =f$ in $B_+$ and $u|_{\partial B_+}=0$.

I want to show that the even extensions of $u,f$ denoted $v,F$ respectively have the following properties. Firstly, that $v \in H_0^1(B)$, and secondly that $v$ is a weak solution of the extended Dirichlet BVP, i.e, $-\Delta v = F$ in $B$ and $v|_{\partial B} = 0$.

I think I should use the fact that the upper half disc is star-shaped but I can't get much further. I approximate $u$ with smooth functions $u_m \in C_0^\infty(B_+)$ and extend these evenly to $\tilde{u}_m$.

What is next? Is this the right approach?

Thanks

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  • $\begingroup$ Are you sure this is true? $\endgroup$ – Giuseppe Negro Jun 5 at 7:46
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This does not seem to be true. Consider the analogous problem in one dimension, where $B$ reduces to $(-1, 1)$ and $B_+=(0,1)$. Let $f=-2$. Then the unique solution to $$ \begin{cases} u''=-2, & (0, 1), \\ u(0)=u(1)=0, \end{cases} $$ is $u(x)=x(1-x)$, and its even extension is $$ v(x)=|x|(1-|x|), $$ which is NOT the solution to $$ \begin{cases} w''=-2, & (-1, 1), \\ w(-1)=w(1)=0, \end{cases}$$ because that solution is $w(x)=1-x^2$.

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