arXiv subjects

Wiktoria Zatoń

Publications and source records attributed to Wiktoria Zatoń.

3 recordsLinked to original sources

arXiv2021

$C^\infty$ partial regularity of the singular set in the obstacle problem

We show that the singular set $Σ$ in the classical obstacle problem can be locally covered by a $C^\infty$ hypersurface, up to an "exceptional" set $E$, which has Hausdorff dimension at most $n-2$ (countable, in the $n=2$ case). Outside this exceptional set, the solution admits a polynomial expansion of arbitrarily large order. We also prove that $Σ\setminus E$ is extremely unstable with respect to monotone perturbations of the boundary datum. We apply this result to the planar Hele-Shaw flow, showing that the free boundary can have singular points for at most countable many times.

math.AP↗

arXiv2021

Runge Approximation and Stability Improvement for a Partial Data Calderón Problem for the Acoustic Helmholtz Equation

In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on \cite{AU04, KU19}. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.

math.AP↗

arXiv2020

Tent space well-posedness for parabolic Cauchy problems with rough coefficients

We study the well-posedness of Cauchy problems on the upper half space $\mathbb{R}^{n+1}_+$ associated to higher order systems $\partial_t u =(-1)^{m+1}\mbox{div}_m A\nabla ^m u$ with bounded measurable and uniformly elliptic coefficients. We address initial data lying in $L^p$ ($1 2$, what is also new for the case $m>1$.

math.AP↗