How can we prove that if $\int_U{f}=0$,then the homogeneous Neumann problem $\Delta u=f$on U,and $\frac{\partial u}{\partial n}=0$ on $\partial U$ has a weak solution in $H^1(U)$?
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Dear Daniel, I think that mathoverflow is not the correct forum for this question unfortunately, since it is a typical homework problem. You could try asking it at, for example, math.stackexchange or artofproblemsolving. However, homework problems copied verbatim without any other text can be frowned upon. Therefor I recommend that you add what you have tried and where you get stuck. 

