arXiv subjects

Nina Uraltseva

Publications and source records attributed to Nina Uraltseva.

2 recordsLinked to original sources

arXiv2024

Properties of the phase boundary in the parabolic problem with hysteresis

We study solutions of parabolic equations with a discontinuous hysteresis operator, described by a free interface boundary. It is established that for spatially transverse initial data from the space $W^{2-2/q}_q$ with $q > 3$, there exists a solution in the space $W^{2,1}_q$, where the interface boundary exhibits Holder continuity with an exponent $1/2$. Furthermore for initial data from the space $W^2_\infty$, it is proven that the interface boundary satisfies the Lipschitz condition. It is shown that for non-transversal initial data, solutions with an interface boundary do not exist.

math.AP↗

arXiv2007

The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions

For the parabolic obstacle-problem-like equation $$Δu - \partial_t u = λ_+ χ_{\{u>0\}} - λ_- χ_{\{u<0\}} ,$$ where $λ_+$ and $λ_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\partial\{u>0\} \cup\partial\{u<0\}$ is in a neighborhood of each ``branch point'' the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper \cite{imrn} to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.

math.AP↗