On regularity properties of solutions to the 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.   We restrict our consideration on the so-called "strong solutions" belonging to the Sobolev class $W^{2,1}_q$ with sufficiently large $q$ and prove that in fact $q=\infty$. In other words, we establish the optimal regularity of solutions. Our arguments are based on quadratic growth estimates for solutions near the free boundary. 
D.E. Apushkinskaya and N.N. Uraltseva
math.AP