Analisi MaTÈmatica allo SBAI
Mercoledì 5 Febbraio
Marta Calanchi (Univ. Di Milano)
Positive eigenvectors and simple nonlinear maps
Abstract: We define (linear) special operators L and (nonlinear) compatible maps P for which the sum F = L − P does not have a point with three preimages. The scheme encapsulates known examples of simple maps (homeo- morphisms, global folds) between Banach spaces and the weaker, geometric, hypotheses suggest new cases: L may be the Laplacian with various boundary conditions, as in the well known Ambrosetti-Prodi theorem, or operators in non-divergence form, as in a recent result of Sirakov et alii. Compatible maps P include the Nemitskii map P(u) = f(u) and may be non-local, even non-variational. The key ingredient is a fruitful extension by Marek of the Krein-Rutman theorem.
This is joint work with Carlos Tomei (PUC-Rio).
Seunghyeok Kim (Hanyang University)
A compactness theorem for the boundary Yamabe problem
Abstract: We concern C^2-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are 4, 5, or 6. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions.
Applying this result and the positive mass theorem, we deduce the C^2-compactness for all 4- and 5-manifolds (which may be non-umbilic).
We also show that the C^2-compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes, complementing the work of Almaraz (2011). This work is collaborated with M. Musso (University of Bath, UK) and J. Wei (UBC, Canada)
Ore 16.20 Tea Break
Federica Sani (Univ. Di Milano)
Title: Asymptotics for a parabolic equation with exponential nonlinearity
Abstract: We consider the Cauchy problem for a 2-space dimensional heat equation with exponential nonlinearity. More precisely, we consider initial data in $H^1(\mathbb R^2)$, and a square-exponential nonlinearity, which is critical in the energy space $H^1(\mathbb R^2)$ in view of the Trudinger-Moser inequality. By means of energy methods, we study the asymptotics of solutions below the ground state energy level. The splitting between blow-up and global existence for low energies is determined by the sign of a suitable functional, and it is related to the corresponding Trudinger-Moser inequality. This is a joint work with Michinori Ishiwata, Bernhard Ruf, and Elide Terraneo.