Ομιλία του κ. Νικόλαου Λαμπρόπουλου (U Patras), στο Σεμινάριο του Τομέα Μαθηματικών της ΣΕΜΦΕ, την Παρασκευή 24 Νοεμβρίου στις 13:35, στην Αίθουσα Σεμιναρίων του τομέα Μαθηματικών (2ος όροφος, κτίριο Ε).
Τίτλος : Study and solvability of nonlinear PDEs in the presence of symmetries.
Περίληψη : In this lecture, we present the most interesting aspects in the study and the solvability of nonlinear of scalar curvature of the generalized type PDEs, (under Dirichlet or Neumann conditions), of upper critical Sobolev exponent on Euclidean domains and on compact Riemannian manifolds with boundary, the data and the functions being invariant under the action of a compact subgroup of the isometry group. In this framework, we calculate the precise values of the best constants in the presented Sobolev type inequalities (i.e. Sobolev, Nash, Hardy, and Hardy-Sobolev inequalities) in the critical of supercritical case without any assumption concerning the “shape” of the boundary (i.e. some convexity) confirming that the symmetry of a domain is an intrinsic property characterizing both the domain itself and its boundary. Subsequently, an alternative method to solving critical PDEs is presented. Finally, we announce some new results concerning the Fujita parabolic equation with supercritical nonlinearity and the Gelfand parabolic equation with supercritical dimension.