  We prove the existence of embedded non planar free boundary minimal disks
into rotationally symmetric ellipsoids of $\mathbb{R}^3$. The construction
relies on the optimization of combinations of first and second Steklov
eigenvalues renormalized by the length of the boundary, among metrics on the
disk. We also prove that non planar free boundary harmonic maps from a disk
into a ellipsoid of $\mathbb{R}^3$ such that the coordinate functions are first
or second Steklov eigenfunctions with respect to the associated critical metric
is a minimal immersion (without branched points) and that if the critical
metric is even with respect to the coordinates of the disk, then the minimal
immersion is an embedding.
