3 research outputs found
Striped periodic minimizers of a two-dimensional model for martensitic phase transitions
In this paper we consider a simplified two-dimensional scalar model for the
formation of mesoscopic domain patterns in martensitic shape-memory alloys at
the interface between a region occupied by the parent (austenite) phase and a
region occupied by the product (martensite) phase, which can occur in two
variants (twins). The model, first proposed by Kohn and Mueller, is defined by
the following functional: where
is periodic in and almost everywhere.
Conti proved that if then the minimal specific
energy scales like ,
as . In the regime , we improve Conti's results, by computing exactly the
minimal energy and by proving that minimizers are periodic one-dimensional
sawtooth functions.Comment: 29 pages, 3 figure
Froth-like minimizers of a non local free energy functional with competing interactions
We investigate the ground and low energy states of a one dimensional non
local free energy functional describing at a mean field level a spin system
with both ferromagnetic and antiferromagnetic interactions. In particular, the
antiferromagnetic interaction is assumed to have a range much larger than the
ferromagnetic one. The competition between these two effects is expected to
lead to the spontaneous emergence of a regular alternation of long intervals on
which the spin profile is magnetized either up or down, with an oscillation
scale intermediate between the range of the ferromagnetic and that of the
antiferromagnetic interaction. In this sense, the optimal or quasi-optimal
profiles are "froth-like": if seen on the scale of the antiferromagnetic
potential they look neutral, but if seen at the microscope they actually
consist of big bubbles of two different phases alternating among each other. In
this paper we prove the validity of this picture, we compute the oscillation
scale of the quasi-optimal profiles and we quantify their distance in norm from
a reference periodic profile. The proof consists of two main steps: we first
coarse grain the system on a scale intermediate between the range of the
ferromagnetic potential and the expected optimal oscillation scale; in this way
we reduce the original functional to an effective "sharp interface" one. Next,
we study the latter by reflection positivity methods, which require as a key
ingredient the exact locality of the short range term. Our proof has the
conceptual interest of combining coarse graining with reflection positivity
methods, an idea that is presumably useful in much more general contexts than
the one studied here.Comment: 38 pages, 2 figure