Scientific seminar on the differential and functional differential equation under the guidance of prof. Sternin B. Yu. and prof. Savin A.Yu

Speaker: P.A. Sipailo (RUDN University)
Topic: On a Sobolev problem with a nonlocal boundary condition.

We consider a Sobolev problem on the 2-torus (and one of the two generating circles is its submanifold) with a nonlocal boundary condition, which involves (a modified) spherical mean operator. The latter operator is expressed as a sum of two Fourier Integral Operators associated with the geodesic flow on the torus (one operator for the "negative" time and one for the "positive" time), enabling us to derive the Fredholm property of our Sobolev problem using the methods of relative elliptic theory and G-theory. This simple example shows how Fourier Integral Operators arise in "daily life" when dealing with differential equations and how one can work with them.