Hello, I really need help with this claim:

If u is harmonic function on connected Lipschitz domain D in R3 vanishing continuously on an open set V contained in the boundary of D and whose normal derivative vanishes on a subset of V with positive surface measure, then u is identically zero on D.

Is it true or not? Note that the domain is Lipschitz , neither C1,1 nor convex.

Thank you so much for your reply !!!