# Example for a weak solution of Poisson's equation on a sphere that is not classical

Let $$M:=S^2$$ be two-dimensional the unit sphere. For a given source function $$f \in H^{-1}(M)$$ I want to find a weak solution $$u \in H^1(M)$$ that is not classical such that for all $$v \in H^1(M)$$ we have $$\int_M \nabla u \cdot \nabla v \, dx= \int_M fv \, dx.$$

Is there any literature where I can find examples for such problems?

• What do you mean by the Sobolev space $H^1_0$ considering your domain is a sphere? – maxmilgram May 5 at 15:46
• Right, it should be $H^1$ I guess since there is no boundary if it is embedded in $\mathbb R^3$ – Tesla May 5 at 19:43
• To obtain an example, why dont you start with $u\in H^1, u\notin C^2$ and then calculate $f$? – maxmilgram May 6 at 6:18
• I can't come up with a $u \in H^1, u \notin C^2$ on the two-dimensional unit sphere. I know that the Laplace-Beltrami-Operator on the two-sphere should be \begin{align} \Delta_{S^2}&= \frac{1}{\sqrt {\det(g)}}\sum_{i=1}^2 \sum_{j=1}^2 \frac{\partial}{\partial X_i}\Bigl(g^{ij}\frac{\partial}{\partial X_j}\sqrt{\det(g)}\Bigr ) \\ &=\frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \bigl(\frac{\partial}{\partial \theta} \sin \theta \bigr) + \frac{1}{\sin ^2 \theta} \frac{\partial^2}{\partial^2 \phi}. \end{align}. Do I need to use it? – Tesla May 7 at 12:49