Henrik Shahgholian: equations with discontinus RHS and free boundaries

Abstract:
In this talk, I shall overview some recent developments of free boundaries coming from
semi linear problems with discontinuous right hand side

Lu=G(x,u),

where L is an elliptic operator, and G, has a discontinuity in its second variable at  u=0.
This produces a so-called free boundary reminiscent of the classical obstacle problem
where G(x,u)  is f(x)H(u) with H being Heaviside function.

I shall present various forms of the function G, where free boundary occurs, and will then also overview the existing results.
For some of the problems I shall explain the ideas, tools and methodology of the theory.