We study the existence of special Lagrangian submanifolds with boundary using area minimization. We prove that in a K\"{a}hler-Einstein manifold,if a smooth Lagrangian surface $\Sigma$ with boundary on a complex hypersurface $M$ is critical for the area functional among all Lagrangian variations which leave $M$ invariant, then $\Sigma$ has zero mean curvature. We also study the boundary regularity for the Lagrangian area-minimizers with respect to the free boundary $M$ and conclude that they are smooth everywhere at the boundry.
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