ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 13, pp. 1734–1763.
Pleiades Publishing, Ltd., 2017.
Boundary Value Problems with Free Surfaces
in the Theory of Phase Transitions
V. G. Osmolovskii
Saint Petersburg State University, St. Petersburg, 199034 Russia
Abstract—The aim of the paper is to show, using the one-dimensional problem as an exam-
ple, what is to be expected and what should be pursued when studying the multidimensional
case. The one-dimensional case has been chosen as a model, because here the problem admits
an explicit solution permitting one to follow the phase transformation process.
The steady-state problem on phase transitions in continuum mechanics can be viewed as
a nonstandard problem of the calculus of variations, which generates equally nonstandard non-
linear boundary value problems for systems of diﬀerential equations (namely, the Euler equations
for the energy functional). We describe the physical statement of this problem in the introduction.
All bibliographic remarks (on the statement of the problem and other topics) are gathered at the
end of the paper.
In the quadratic approximation, the free energy density of a single-phase inhomogeneous aniso-
tropic medium occupying a domain Ω ⊂ R
, m =1, 2, 3, can be written as
F (∇u, t
(∇u) − ζ
(∇u) − ζ
) − t
(∇u) − ζ
where u = u(x), x ∈ Ω, is the displacement ﬁeld, (∇u)
strain tensor, ζ
(x) is the residual strain tensor, t
(x) is the temperature deviation from
a given value, and F
) is a second-order polynomial of the argument t
. The functions a
and the coeﬃcients of the polynomial F
depend on x ∈ Ω. They are determined by the elastic
and thermodynamic characteristics of the medium and are subject to traditional constraints, and
summation from 1 to m over repeated indices is assumed in the formulas.
Let g and f be the bulk and surface force ﬁelds acting on the elastic medium. Then the strain
energy functional for the density (0.1) is given by the formula
F (∇u, t
,x) dx −
g · udx−
f · udS, (0.2)
and the equilibrium displacement ﬁeld ˆu for a given temperature distribution is a solution of the
], ˆu ∈
is the set of admissible displacement ﬁelds, which is speciﬁed by the boundary value of
the function u on part of the boundary of Ω (possibly empty or coinciding with the entire ∂Ω).
Deformation processes in two-phase elastic media are accompanied with phase transitions in-
volving changes in the crystal structure. We assume that only two structures can be realized for
two-phase media. We mark them with the symbols + and −, and the corresponding variables in
the representations (0.1) of the energy densities F
,x) are denoted by a