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We show that for any continuous $f: (\\Sigma, \\partial \\Sigma) \\rightarrow (N, M)$ for which the induced homomorphism on certain fundamental groups is injective, there exists a branched minimal immersion of $\\Sigma$ solving the free boundary problem $(\\Sigma, \\partial \\Sigma) \\rightarrow (N, M)$, and minimizing area among all maps which induce the same action on the fundamental groups as f. 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Let $\\Sigma$ be a compact orientable surface with boundary. We show that for any continuous $f: (\\Sigma, \\partial \\Sigma) \\rightarrow (N, M)$ for which the induced homomorphism on certain fundamental groups is injective, there exists a branched minimal immersion of $\\Sigma$ solving the free boundary problem $(\\Sigma, \\partial \\Sigma) \\rightarrow (N, M)$, and minimizing area among all maps which induce the same action on the fundamental groups as f. 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