Dissertation committee:
Π. Γιαννιώτης, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ,
Ν. Καραχάλιος, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Θεσσαλίας,
Γ. Μπαρμπάτης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ,
Π. Σμυρνέλης, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ,
Ι. Στρατής, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ,
Α. Τερτίκας, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Κρήτης,
Ε. Φίλιππας, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Κρήτης.
Summary:
The aim of this doctoral dissertation is to investigate the validity and additional properties of Hardy's well known inequality in various settings beyond the Euclidean. The dissertation consists of four chapters. Chapter 1 offers background on Hardy inequalities, particularly so in the non-Euclidean setting. In Chapter 2, we introduce a method of integration along integral curves to obtain Hardy inequalities for the first order differential operator X in a given manifold M with volume form ω. Chapter 3 is concerned with higher order Rellich inequalities related to general elliptic operators with constant coeficients, other than the classic polyharmonic operator. In this case, we show that a Rellich inequality can be expressed in terms of an induced Finsler distance which is given in terms of the symbol of the operator. This new type of inequality is shown to be
sharp in the case where the underlying domain is a half-space and the symbol satisfies a convexity condition, while comparisons are made for the case of a convex domain, yielding results that are superior to those obtained by more crude methods, in specific situations. Finally, in Chapter 4, we deal with the sensitivity of the Hardy constant under perturbations of the domain in the case where the distance is measured from a boundary submanifold. Specifcally, we find the Hardy constant to be both continuous and differentiable (in the Gateaux sence) under such perturbations, assuming some regularity conditions on the boundary.
