Unit:
Department of MathematicsLibrary of the School of Science
Author:
Branikas Panagiotis-Christos
Dissertation committee:
Γεράσιμος Μπαρμπάτης, Καθηγητής, Τμήμα Μαθηματικών, Ε.Κ.Π.Α.
Ιωάννης Γιαννούλης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Ιωαννίνων
Ευαγγελία Κόττα - Αθανασιάδου, Αναπληρώτρια Καθηγήτρια, Τμήμα Μαθηματικών, Ε.Κ.Π.Α.
Παναγιώτης Σμυρνέλης, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, Ε.Κ.Π.Α.
Ιωάννης Στρατής, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, Ε.Κ.Π.Α.
Αχιλλέας Τερτίκας, Καθηγητής, Τμήμα Μαθηματικών και Εφαρμοσμένων Μαθηματικών, Πανεπιστήμιο Κρήτης
Ευστάθιος Φίλιππας, Καθηγητής, Τμήμα Μαθηματικών και Εφαρμοσμένων Μαθηματικών, Πανεπιστήμιο Κρήτης
Original Title:
Εκτιμήσεις πυρήνων θερμότητας για ελλειπτικούς διαφορικούς τελεστές τέταρτης τάξης
Translated title:
Heat kernel estimates for fourth-order elliptic operators
Summary:
In this doctoral dissertation, entitled "Heat kernel estimates for fourth-order elliptic operators" we consider a class of fourth order uniformly elliptic operators in planar Euclidean domains and study the associated heat kernel. For operators with L^∞ coefficients we obtain Gaussian estimates with best constants, while for operators with constant coefficients we obtain short time asymptotic estimates. The novelty of this thesis is that we do not assume that the associated symbol is strongly convex. The short time asymptotics reveal a behavior which is qualitatively different from that of the strongly convex case.
We also obtain heat kernel estimates for a class of fourth order non-uniformly elliptic operators in two dimensions. Contrary to existing results, the operators considered have symbols that are not strongly convex. This rises certain difficulties as it is known that, as opposed to the strongly convex case, there is no absolute exponential constant. Our estimates involve sharp constants and Finsler-type distances that are induced by the operator symbol. The main result is based on two general hypotheses, a weighted Sobolev inequalitry and an interpolation inequality, which are related to the singularity or degeneracy of the coefficients.
Main subject category:
Science
Keywords:
heat kernels, fourth-order elliptic operators, Gaussian estimates, asymptotic estimates, strongly convex, Finsler metric, steepest descent method
