# Minimum is attained in a subset of a Sobolev space

Let $\Omega \subset \mathbb R^n$. I have a functional of the form,

$$\int_{\Omega}f(x,u,\nabla u)dx$$ where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian manifold and $$f(x,s,z) = g(x,s,z) + |z-\mathbb P_s(z)|^p.$$ The mapping $\mathbb P_s$ is the projection into the tangent space of $M$ at $s$. The function $g$ satisfies $a|z|^p \leq g(x,s,z) \leq b(1+|z|^p)$ and is sufficiently smooth for all of the objects above to exist and be well defined.

Question: Given $u_0 \in W^{1,p}(\Omega, M)$, can we say that

$$\inf_{u+u_0 \in W^{1,p}(\Omega, M)}\int_{\Omega}f(x,u,\nabla u)dx = \inf_{u+u_0 \in W^{1,p}(\Omega, \mathbb R^d)}\int_{\Omega}f(x,u,\nabla u)dx?$$

I.e. given boundary data in the manifold, the minimiser must remain in the manifold? If not then how else could I penalise $g$ to ensure this property but still have a $p-$growth condition in $z$.

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