Supervisors info:
Γεράσιμος Μπαρμπάτης Αναπλ. Καθηγητής ΕΚΠΑ (Επιβλέπων), Νικόλαος Αλικάκος Καθηγητής ΕΚΠΑ, Ιωάννης Στρατής Καθηγητής ΕΚΠΑ
Summary:
In Chapter 1 we first define the Sobolev Spaces W^{k,p}(U) when k a non
-negative integer and p greater than or equal to 1.Then various properties of
weak derivatives follow. We then define the fractional Sobolev Spaces
W^{k,p}(U) , for k belonging in the interval (0,1), and p being greater than or
equal to 1. Imbedding theorems for these spaces are also proved. We then
proceed to define the fractional Sobolev spaces W^{k,p}(U) with k being a
noninteger number, greater than or equal to 1, p being greater than or equal to
1 , and then the imbedding theorem for these Sobolev spaces follows.
We then give the definition of the standard Laplace operator Δ and subsequantly
we define the fractional Laplace operator (-Δ)^k . This is followed by various
properties of the fractional Laplace operator. Chapter 1 closes with an
alternative definition of the Sobolev Space H^k(R^n) , which makes use of the
Fourier transform, with k being positive or, respectively, negative. In Chapter
2 we first define the distance function d(x) and we study its main
properties. Then follows the standard Hardy inequality for the Laplace operator
on a bounded convex domain with C^2 boundary. In Chapter 3 we study Hardy
inequalities for the fractional Laplace operator. We first present a Hardy
inequality with best constant for the fractional Laplacian (-Δ )^{α/2} when Ω
is a bounded domain with non-empty boundary and α belongs in the interval
(1,2). In this case the right -hand side of the inequality does not involve the
distance function d(x) but a more complex expression M_α(x). In the special
case where
Ω is bounded and convex domain , this implies the Hardy inequality with the
function d(x) , and best constant.
Keywords:
Inequalities, Hardy, Fractional, Laplacian, Sobolev
