# dg.differential geometry – Dirichlet problem for minimizing harmonic maps

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 $$N$$.