| Abstract |
In this dissertation, we focus on phase transition type problems. Let us briefly
discuss the physical motivation of such problems in the simple case of two phases.
We are given some substance in a container, which may exhibit two phases, say
a1 and a2, and we would like to describe it mathematically. One approach could
be that the interface formulation is driven by a variational principle, that is the
pattern in the outcome of the minimization of a certain energy. For this, we may
consider a “double well” potential W such that W(a1) = W(a2) = 0 and W > 0
otherwise. Next, one introduces a gradient term that penalizes the formulation of
interfaces and measures interface energy. This is the Van der Waals free energy
functional. To be more precise,
Jε(u) = Z
Ω
Å
ε
2
|∇u|
2 +
1
ε
W(u)
ã
dx , u : Ω ⊂ R
n → R. (1.1)
We often call Jε as the Allen-Cahn functional. Such a gradient term reduces
the number of interfaces of the minimizers of Jε, which turn out to be smooth
functions interpolating between the phases a1 and a2 with level sets approaching
hypersurfaces of least possible area. Therefore, our problem is also closely related
to the theory of minimal surfaces. Phase transition type problems arise in many
experimental disciplines, such as material science.
For studying three or more phases, one naturally is lead to the vector case. The
equations arising in phase transition type problems are called Allen-Cahn equations
or systems of equations in the case of more than two phases. Particularly,
∆u = Wu(u) , where u : Ω ⊂ R
n → R
m. (1.2)
This dissertation is consisted of five independent parts. In the first part
contained in Chapter 2, which is a work together with professor N. Alikakos and
professor A. Zarnescu that can be found in [1], we study entire solutions of the
Allen-Cahn systems that are also minimizers. In particular, the specific feature
of our systems are potentials having a finite number of global minima (i.e. the
phases), with sub-quadratic behavior locally near their minima. We focus on
qualitative aspects and we show the existence of entire solutions in an equivariant
setting connecting the minima of W at infinity, thus modeling many coexisting
phases, possessing free boundaries and minimizing energy in the symmetry class.
The existence of a free boundary can be related to the existence of a specific
sub-quadratic feature, a dead core, whose size is also quantified.
In the second part which is in Chapter 3, motivated by the relationship of phase
transition type problems with minimal surfaces, we determine a transformation
that transforms equipartitioned solutions of the Allen-Cahn equations in dimension
three to the minimal surface equation of one dimension less. This is an application
of a more general transformation introduced in this work which relates the solutions
of the Allen-Cahn equations that are equipartitioned to solutions of the incompressible
Euler equations with constant pressure. Other applications are De Giorgi type
results, that is, the level sets of entire solutions are hyperplanes. Also, we determine
the structure of solutions of the Allen-Cahn system in two dimensions that satisfy
the equipartition of the energy and we apply the Leray projection to provide
explicit entire solutions to analyze this structure. In addition, we obtain some
examples of smooth entire solutions of the Euler equations, some of which, can
be extended to the Navier-Stokes equations for specific type of initial conditions.
This work can be found in [2].
In the following part, which concerns the work in [3] and is contained in
Chapter 4, we are dealing with the Γ− convergence of the Allen-Cahn functional
with Dirichlet boundary conditions in the vectorial case. Let us briefly describe
the analog of the Γ−limit result in the scalar case. Assume Fε is the ε−energy
functional of the Allen-Cahn equation,
Fε(u, Ω) := ®R
Ω
ε
2
|∇u|
2 +
1
εW(u)dx , u ∈ W1,2
(Ω; R)
+∞ , elsewhere
(1.3)
then it is a classical well known theorem that the Γ−limit of Fε is the perimeter
functional F0 which measures the transitions between the two phases of the problem,
i.e.
F0(u, Ω) := ®
σHn−1
(Su) , u ∈ BV (Ω; {−1, 1})
+∞ , elsewhere
(1.4)
where W : R → [0, +∞) , {W = 0} = {−1, 1} , σ =
Z 1
−1
»
2W(u)du (1.5)
and Su is the singular set of the function u. (1.6) Thus, the interfaces of the limiting problem will be minimal surfaces. We provide
all the appropriate references and previous fundamental contributions on the topic
in this third part. So, in the vectorial case that we study, one expects that the
Γ−limit turn out to be the perimeter functional that measures the transition
between the N− phases of the problem. We prove this fact with the constraint
of boundary conditions. In this case, the minimizers of the limiting functional are
closely related to minimizing partitions of the domain. Moreover, utilizing that
the triod and the straight line are the only minimal cones in the plane together
with regularity results for minimal curves, we determine the precise structure of
the minimizers of the limiting functional, and thus the limit of minimizers of the
ε− energy functional as ε → 0. We also prove that the minimizer of the limiting
functional in the disc is unique.
Next, in the forth part which is in Chapter 5, we study fully nonlinear elliptic
equations via the notion of P− functions. P− functions can be thought as
quantities of the solution of a general fully nonlinear partial differential equation
that satisfy the maximum principle. Perhaps the most well-known example is
P(u; x) = 1
2
|∇u|
2 − W(u) (1.7)
that is related to the Allen-Cahn equation
∆u = W′
(u) , u : Ω ⊂ R
n → R (1.8)
and one important application is the Modica inequality
1
2
|∇u|
2 ≤ W(u) (1.9)
for every bounded solution of (1.8). There are many generalizations to Quasi-linear
elliptic equations among other types of equations with applications such as gradient
bounds and Liouville theorems which we refer in detail in the respective chapter
of this forth part. In our work, which can be found in [4], we introduce the notion
of P− functions for fully nonlinear equations and establish some general criterion
for obtaining such quantities for this class of equations. Some applications are
gradient bounds, De Giorgi-type properties of entire solutions and rigidity results.
Furthermore, we prove Harnack-type inequalities and local pointwise estimates for
the gradient of solutions to fully nonlinear elliptic equations. Additionally, we
consider P−functions for higher order nonlinear equations and for equations of
order greater than two we obtain Liouville-type theorems and pointwise estimates
for the Laplacian.
Finally, in Chapter 6, the last part of our thesis includes a work with professor
C. Makridakis and G. Gkanis that can be found in [5], in which we study applications f Physics Informed Neural Networks to partial differential equations. Physics
Informed Neural Networks is a numerical method which uses neural networks to
approximate solutions of partial differential equations. It has received a lot of
attention and is currently used in numerous physical and engineering problems.
The mathematical understanding of these methods is limited, and in particular, it
seems that, a consistent notion of stability is missing. Towards addressing this issue
we consider model problems of partial differential equations, namely linear elliptic
and parabolic PDEs. We consider problems with different stability properties, and
problems with time discrete training. Motivated by tools of nonlinear calculus of
variations we systematically show that coercivity of the energies and associated
compactness provide the right framework for stability. For time discrete training
we show that if these properties fail to hold then methods may become unstable.
Furthermore, using tools of Γ−convergence we provide new convergence results for
weak solutions by only requiring that the neural network spaces are chosen to have
suitable approximation properties. These techniques can be extended to various
other, possibly nonlinear, problems
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