Date/Time Date(s) - 03/10/202211:30 am - 12:30 pm
Speaker: Sullivan MacDonald (McMaster University)
Title: Sum of squares decompositions with applications to regularity theoryAbstract: Given a non-negative function $f$ belonging to $C^{k,alpha}(mathbb{R}^n)$, it is an open question whether $f$ can be written as a sum of squares of functions that inherit `half’ the regularity of $f$, in the sense that each root function belongs to $C^{k/2,alpha/2}(mathbb{R}^n)$. Recent work shows that such decompositions always exist for $k<4$, while non-decomposable functions exist in $C^{k,alpha}(mathbb{R}^n)$ for $kgeq 4$.
In this talk we prove that functions in lower-order Holder spaces can be decomposed, and discuss possible extensions of this result in higher-order spaces. There are several applications of these decomposition theorems to PDEs, which include: (i) proving hypoellipticity for a class of (pseudo)-differential operators, and (ii) proving the existence of convex solutions to Dirichlet problems for the Monge-Ampere equation. These results are briefly discussed, along with some open problems that are interesting both theoretically and practically.
Location: Hamilton Hall 403