Pratyush Potu will give a talk on 10/14/2025.

The divergence equation ($\mathrm{div}\, u = f$) is a simple PDE with many applications. An explicit solution can be obtained by the so-called Bogovskiĭ operator first introduced in 1979. A key aspect of Bogovskiĭ’s operator is that the resulting solution will satisfy Dirichlet boundary conditions and be bounded in the $W^{1,p}$ norm. In fact, the Bogovskiĭ operator can be generalized to invert any differential operator in the de-Rham complex in arbitrary dimension while satisfying boundary conditions. Moreover, this generalized Bogovskiĭ operator is a pseudodifferential operator of order -1 (essentially a bounded operator on a wide range of function spaces). However, the Bogovskiĭ operator relies on the assumption that the domain under consideration is star-shaped with respect to a ball. Hence, a modified Bogovskiĭ operator was introduced by Acosta, Durán, and Muschietti for the divergence which allows for all the same properties as the original construction but now on a much more general class of domains known as John domains. In this talk, I will discuss these previous results and conclude with my ongoing work on combining these constructions.