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Interfaces and Free Boundaries


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Volume 17, Issue 1, 2015, pp. 93–115
DOI: 10.4171/IFB/335

Published online: 2015-05-26

On regularity properties of solutions to the hysteresis-type problem

Darya E. Apushkinskaya[1] and Nina N. Uraltseva[2]

(1) Universität des Saarlandes, Saarbrücken, Germany
(2) St. Petersburg State University, Russian Federation

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.

Keywords: Free boundary, hysteresis, sub-caloric functions, monotonicity formula, quadratic growth estimates

Apushkinskaya Darya, Uraltseva Nina: On regularity properties of solutions to the hysteresis-type problem. Interfaces Free Bound. 17 (2015), 93-115. doi: 10.4171/IFB/335