Supervisors info:
Μπαρμπάτης Γεράσιμος Καθηγητής Τμήμα Μαθηματικών ΕΚΠΑ,
Στρατής Ιωάννης Καθηγητής, Τμήμα Μαθηματικών ΕΚΠΑ,
Τύρος Κώνσταντίνος Αναπλ. Καθηγητής Τμήμα Μαθηματικών ΕΚΠΑ
Summary:
In this thesis we focus on improved to $L^{p}$ Hardy inequalities with best constants.
First of all, in Chapter 2 we state basic concepts such as (Sobolev Spaces) and important Theorems like (Coarea Formula) ,
which we use them like tools to help in our proofs. In Chapter 3 , we proved The Classical Hardy Inequalities in the simple cases
when $p=2$. The first case is about distance from a point and the other one is distance from the boundary.
In both cases we show that the constants are the optimal.
In Chapter 4 , firstly we study the main geometric assumption concerning $K,\Omega$ and through
a Lemma we give an equivalent formulation. Next, we proved the improved $L^{p}$ Hardy inequalities for any $p>1$
and general case for the distance function $d(x)$ .
We distinct two cases, one is when $p\neq k$ and the other is when $p=k$, where $k$ is the codimension of the smooth surface $K$,
which we consider the distance of $x\in\Omega$. In last section of this chapter,
we prove the optimality of the constants which appearing in $L^{p}$ improved Hardy inequalities.
