On regularity properties of solutions to hysteresis-type problems

We consider equations with the simplest hysteresis operator at

the right-hand side. Such equations describe the so-called processes "with

memory" in which various substances interact according to the hysteresis

law. The main feature of this problem is that the operator at the

right-hand side is a multivalued.

We present some results concerning the optimal regularity of solutions.

Our arguments are based on quadratic growth estimates for solutions near

the free boundary. The talk is based on joint work with Darya

Apushkinskaya.