I really need your help. I have the following problem:
Let Omega be connected Lipschitz domain. The boundary of Omega is Gamma. It has two disjoint smooth open subsets Gamma_1 and Gamma_2. Gamma_i (i=1,2) is of class C^{1,1}.
Let us defined space:

V= { v in H1(omega)^3 | div v = 0 in Omega, v= 0 on Gamma_1, v x n = 0 on Gamma_2}

I want to proof thic implication:

If v in V and curl v = 0 in Omega, then v = 0 in Omega.

So far, I have introduce to Omega smooth cuts to make simply connected domain Omega0. Since v is rotation-free function, there exist unique class q in H1(Omega0)/R such that: v = grad q in Omega 0
Since v is divergence - free, laplace q = 0 in Omega0.

What should be the next step to get q = 0 in Omega0. This would imply that v=0 in Omega.

I really need your help. Thank you so much!