On regularity properties of solutions to the hysteresis-type problems

TL;DR

This paper proves the optimal regularity of solutions to hysteresis-type equations with memory effects, showing that strong solutions in Sobolev spaces are actually infinitely smooth under certain conditions.

Contribution

It establishes the optimal regularity of strong solutions to hysteresis equations, demonstrating they belong to the class $W^{2,1}_q$ with $q=\infty$, which was previously unknown.

Findings

Solutions are in $W^{2,1}_q$ for all $q$, including $q=\infty$

Quadratic growth estimates near the free boundary are key to the proof

Regularity results apply to equations modeling processes with hysteresis law

Abstract

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.