arXiv subjects

N. N. Uraltseva

Publications and source records attributed to N. N. Uraltseva.

3 recordsLinked to original sources

arXiv2014

Free boundaries in problems with hysteresis

In this note we present a survey concerning parabolic free boundary problems involving a discontinuous hysteresis operator. Such problems describe biological and chemical processes "with memory" in which various substances interact according to hysteresis law. Our main objective is to discuss the structure of the free boundaries and the properties of the so-called "strong solutions" belonging to the anisotropic Sobolev class $W^{2,1}_q$ with sufficiently large $q$. Several open problems in this direction are proposed as well.

math.AP↗

arXiv2014

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.

math.AP↗

arXiv2012

Uniform estimates near the initial state for solutions to the two-phase parabolic problem

This paper is devoted to a proof of optimal regularity, near the initial state, for weak solutions to the two-phase parabolic obstacle problem. The approach used here is general enough to allow us to consider the initial data belonging to the class $C^{1,1}$.

math.AP↗