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?

  • $\begingroup$ What do you mean by the Sobolev space $H^1_0$ considering your domain is a sphere? $\endgroup$ – maxmilgram May 5 at 15:46
  • $\begingroup$ Right, it should be $H^1$ I guess since there is no boundary if it is embedded in $\mathbb R^3$ $\endgroup$ – Tesla May 5 at 19:43
  • $\begingroup$ To obtain an example, why dont you start with $u\in H^1, u\notin C^2$ and then calculate $f$? $\endgroup$ – maxmilgram May 6 at 6:18
  • $\begingroup$ 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? $\endgroup$ – Tesla May 7 at 12:49

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