On the strong locality of differential operators

It is well known that, given a Sobolev function vanishing on a measurable set, the gradient must vanish almost everywhere on that set. This property is usually called “locality of the gradient operator.” In the seminar, we will introduce the notion of locality for general linear (first-order) differential operators and we will discuss some sufficient and necessary conditions for locality to hold. We will present several examples and, if time allows, a complete catalogue of differential operators in the 2D setting. This is part of ongoing projects with G. Alberti (Pisa) and G. Del Nin (MPI Leipzig).