The HADES seminar on Tuesday, March 4th, will be at 3:30pm in Room 740.

Speaker: Yuchen Mao

Abstract: Solving an underdetermined PDE such as a divergence equation plays a central role in problems like general relativistic gluing. Starting from divergence equations on Euclidean spaces, I will introduce a method of constructing integral solution operators for a wide class of underdetermined differential operators with prescribed support properties. By duality, this will also produce integral representation formulas for overdetermined differential operators. The method extends various ideas from Bogovskii, Oh-Tataru, and Reshetnyak. The construction is based on an assumption called the recovery on curves condition (RC) imposed on the operators. I will also give an algebraic sufficient condition of RC that is easier to verify, which is called the finite-codimensional cokernel condition (FC). At the end, I will show some examples that satisfy FC on space forms and derive their integral formulas in the flat case. This is joint work with Philip Isett, Sung-Jin Oh, and Zhongkai Tao.