Let $(M,g_1)$ and $(N,g_2)$ be smooth Riemannian manifolds such that $M$ is compact with smooth boundary and $N$ is complete.

Given a map $phi in W^{1,2}(M,N)$, can we find a minimizing harmonic map $u in W^{1,2}(M,N)$ such that $u=phi$ on $partial M$ in the sense of trace?

If $N$ is compact, the existence follows from a minimizing sequence with its weak limit. I am not sure if the same conclusion holds for noncompact $N$.