## On the regularity of solutions of free boundary problems

##### Abstract

Optimal design problems have aroused particular interest in the scientific com-
munity over the past thirty years. In physics, for example, they find application
in the investigation of the minimal energy configurations of a mixture of two
materials in a bounded and connected open set.
The fascination of such problems derives from their variational formula-
tion, which involves not only the state function of a system, but also a shape,
that is a set. If a penalizing contribution of perimeter form, due to a surface
energy, is added to the integral mass energy, dependent on the configuration
state-shape, the problem becomes even more intriguing and inspiring.
It is not straightforward to investigate the regularity of minimizing pairs
because the two energies have different dimensions under commong scalings:
once a homothety of factor r is applied, the first energy “behaves” as a vol-
ume (rescaling with factor r
n
), the second as a perimeter (rescaling with factor
r
n1
). The coexistence of the two types of energies is managed using techniques
and tools of both the Calculus of Variations and the Geometric Measure The-
ory.
In the first part of this thesis we deal with two optimal design problems,
in which the integral functions that constitute the mass energy have different
growths.
If their growth is at most quadratic, we prove the C
1,μ regularity of the
interface of the shape that constitutes the optimal pair, up to a singular set of
Hausdorff dimension less than n 1. The technique used combines the regu-
larity theories of the Λ-minimizers of the perimeter and the minimizers of the
Mumford-Shah functional.
If the integrands have at most a polynomial growth of degree p, the anal-
ysis becomes more involved. The C
1,μ regularity of the interface remains an
open problem. However, it is proved that the optimal shape of the problem
is equivalent to an open set with a topological boundary that differs from its
reduced boundary for a set of Hausdorff dimension less than or equal to n 1.
In the second part of the thesis we address to a completely different varia-
tional problem, involving a frustrated spin system on a (one-dimensional and
two-dimensional) lattice confined in two magnetic anisotropy circles.
This topic is of significant scientific interest, as it is useful for understand-
ing the behavior of low-dimensional magnetic structures existing in nature.
The frustration parameter α ¡ 0 of the system averages the ferromagnetic
and antiferromagnetic interactions that coexist in the energy. The minimal
energy state of the system, for α ¤ 4, consists of a spin that “lives” within
only one of the two magnetic anisotropy circles and has a positive or negative
chirality.
We find the correct rescaling of the functional and prove the energy needed
to detect the two phenomena that break the rigid minimal symmetry described.
These are chirality transitions and magnetic anisotropy transitions of the spin. [edited by Author]