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In this paper we analyze the singular set in the Stefan problem and prove the following results: - The singular set has parabolic Hausdorff dimension at most $n-1$. - The solution admits a $C^infty$-expansion at all singular points, up to a set of parabolic Hausdorff dimension at most $n-2$. - In $mathbb R^3$, the free boundary is smooth for almost every time $t$, and the set of singular times $mathcal Ssubset mathbb R$ has Hausdorff dimension at most $1/2$. These results provide us with a refined understanding of the Stefan problems singularities and answer some long-standing open questions in the field.
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We revisit and sharpen the results from our previous work, where we investigated the regularity of the singular set of the free boundary in the nonlinear obstacle problem. As in the work of Figalli-Serra on the classical obstacle problem, we show that each stratum can be further decomposed into a `good part and an `anomalous part, where the former is covered by $C^{1,1-}$ manifolds, and the latter is of lower dimension.
We develop a high-order energy method to prove asymptotic stability of flat steady surfaces for the Stefan problem with surface tension - also known as the Stefan problem with Gibbs-Thomson correction.
We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, $minbigl{(-Delta)^su,,u-varphibigr}=0$ in $mathbb R^n$, for general obstacles $varphi$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in cite{GP} to all $sin(0,1)$.
We provide perturbative estimates for the one-phase Stefan free boundary problem and obtain the regularity of flat free boundaries via a linearization technique in the spirit of the elliptic counterpart established by the first author.
We consider the nonlinear Stefan problem $$ left { begin{array} {ll} -d Delta u=a u-b u^2 ;; & mbox{for } x in Omega (t), ; t>0, u=0 mbox{ and } u_t=mu| abla_x u |^2 ;;&mbox{for } x in partialOmega (t), ; t>0, u(0,x)=u_0 (x) ;; & mbox{for } x in Omega_0, end{array}right. $$ where $Omega(0)=Omega_0$ is an unbounded smooth domain in $mathbb R^N$, $u_0>0$ in $Omega_0$ and $u_0$ vanishes on $partialOmega_0$. When $Omega_0$ is bounded, the long-time behavior of this problem has been rather well-understood by cite{DG1,DG2,DLZ, DMW}. Here we reveal some interesting different behavior for certain unbounded $Omega_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $Omega_0$.
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