Trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ invertible on $\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$

Question

It is apparently true that the trace map $\gamma:H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$, is invertible when restricted to the domain $$\{ u \in H^1(\Omega) \mid \Delta u =0 \}.$$

I've been wondering why all morning but I've had no luck. OK, one can set up the well-posed problem $$\Delta u = 0$$ $$\gamma(u) = v$$ but I see no reason why I cannot have a $\Delta u = f \in H^{-1}$ instead.