# Extending harmonic function from half-space to whole space

$$u$$ is harmonic in $$\mathbb{R}^n_+$$ and $$u=0$$ on the boundary. I wish to extend $$u$$ to a harmonic function on $$\mathbb{R}^n$$. Suppose I defined $$u(x_1,...,x_{n-1},x_n)=u(x_1,...,x_{n-1},-x_n)$$ for $$x_n\leq0$$. Clearly $$u$$ would be harmonic on the lower half space. But what about the boundary? If $$u$$ is harmonic at a point $$\xi$$ on the boundary it must satisfy the mean value property i.e. $$0=u(\xi)=\frac{1}{w_n}\int_{|x|=1}u(\xi+rx)dS_x$$ But for that to hold, shouldn't I define $$u(x_1,...,x_{n-1},x_n)=-u(x_1,...,x_{n-1},-x_n)$$?

• Your reasoning seems sound to me. – DisintegratingByParts Nov 11 '19 at 17:51
• @DisintegratingByParts I have seen people go with the first extension rather than the second one. Why is that? – Hrit Roy Nov 14 '19 at 13:02