Abstract
The present thesis is a result of research done in the field of optimizationproblems related to non-local partial differential equations in past three
years. More precisely it focuses on optimal rearrangement problems in the
context of (non-local) fractional Laplace operators.
In various problems in physics, fluid mechanics and economics certain
functions belong to the same rearrangement class (see Section 1.1). Then
one is interested in maximization or minimization of particular energies and
analyzing the properties of the corresponding optimal solutions. This field of
mathematics is based on research of Geoffrey Burton and his collaborators
and students in 80s and 90s (see [13], [16], [17], [14], [15]). Later the results
have been generalized for the p−Laplace operators (see [43], [33], [42], [31]),
biharmonic operator (see [35], [22]), as well as constrained cases (see [44],
[32]).
Most of the classical results known for the Laplace operators have been
generalized in the non-local setting for the first time in this work. Same
time we show that the non-locality implies new phenomena not observed for
local operators. Similarly to the results obtained in [44], the solution of the
energy minimization problem we obtain is not a characteristic function (a
bang-bang function).
Another important feature is the connection between optimal rearrangement problems and free boundary problems, particularly the obstacle problem, known for the classical (local) optimal rearrangement problems. Our
analysis allows to derive the fractional version of the so-called normalized
obstacle problem from the rearrangement context, and obtain a new type of
equation for its solutions. The results in Chapter 3 and Chapter 4 of this
thesis are published in [8] and [7] respectively.
Short about the structure of the thesis: in Chapter 1 we introduce some
backgrounds and applications; in Chapter 2 we introduce some preliminaries
in fractional setting; in Chapter 3, we study the fractional optimal maximization problem; in Chapter 4, we study the fractional optimal minimization
problem; In Chapter 5, we study the fractional analogue of the variational
minimization problem introduced in [37].
| Date of Award | Jul 2022 |
|---|---|
| Original language | English |
| Awarding Institution |
|
| Supervisor | Hayk Mikayelyan (Supervisor), Behrouz Emamizadeh (Supervisor) & Matthias Kurzke (Supervisor) |