Extending Poisson equation from half unit disc

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

• Are you sure this is true? – Giuseppe Negro Jun 5 at 7:46

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 $$w(x)=1-x^2$$.