  In this paper, we establish a min-max theory for constructing minimal disks
with free boundary in any closed Riemannian manifold. The main result is an
effective version of the partial Morse theory for minimal disks with free
boundary established by Fraser. Our theory also includes as a special case the
min-max theory for Plateau problem of minimal disks, which can be used to
generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces
in $\mathbf{R}^n$ to the Riemannian setting.
  More precisely, we generalize the min-max construction of minimal surfaces
using harmonic replacement introduced by Colding and Minicozzi to the free
boundary setting. As a key ingredient to this construction, we show an energy
convexity for weakly harmonic maps with mixed Dirichlet and free boundaries
from the half unit $2$-disk in $\mathbf{R}^2$ into any closed Riemannian
manifold, which in particular yields the uniqueness of such weakly harmonic
maps. This is a free boundary analogue of the energy convexity and uniqueness
for weakly harmonic maps with Dirichlet boundary on the unit $2$-disk proved by
Colding and Minicozzi.
