  We consider the problem of optimal partition of a domain with respect to the
sum of the principal eigenvalues and we prove for the first time regularity
results for the free interface up to fixed boundary. All our results are
quantitative and, in particular, we obtain fine estimates on the continuity of
the solutions and the oscillation of the free interface (in terms of the
modulus of continuity of the normal vector of the fixed boundary), even in the
case of domains with low (Dini-type) regularity. Our analysis is based on an
Almgren-type monotonicity formula at boundary points and an epiperimetric
inequality at points of low frequency, which, together, yield an explicit rate
of convergence for blow-up sequences and the boundary strong unique
continuation property. Exploiting our quantitative blow-up analysis, we manage
to prove clean-up results near one-phase and two-phase points. We define the
notion of free interface inside the fixed boundary, and we prove that the
subset of points of minimal frequency is regular and that the interior free
interface is approaching the boundary orthogonally in a smooth way.
