Non planar free boundary minimal disks into ellipsoids

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.