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Register a new userHigher Regularity of the Free Boundary in the Obstacle Problem for the Fractional Laplacian
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We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.
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The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump Levy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when $s>frac12$, the free boundary is a $C^{1,alpha}$ graph in $x$ and $t$ near any regular free boundary point $(x_0,t_0)in partial{u>varphi}$. Furthermore, we also prove that solutions $u$ are $C^{1+s}$ in $x$ and $t$ near such points, with a precise expansion of the form [u(x,t)-varphi(x)=c_0bigl((x-x_0)cdot e+a(t-t_0)bigr)_+^{1+s}+obigl(|x-x_0|^{1+s+alpha}+ |t-t_0|^{1+s+alpha}bigr),] with $c_0>0$, $ein mathbb{S}^{n-1}$, and $a>0$.
We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $minbigl{-Delta_p u,,u-varphibigr}=0$ in $Omegasubsetmathbb R^n$. Here, $Delta_p u=textrm{div}bigl(| abla u|^{p-2} abla ubigr)$, and $pin(1,2)cup(2,infty)$. Near those free boundary points where $ abla varphi eq0$, the operator $Delta_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $ abla varphi=0$ then $Delta_p$ is singular or degenerate, and nothing was known about the regularity of the free boundary at those points. Here we study the regularity of the free boundary where $ abla varphi=0$. On the one hand, for every $p eq2$ we construct explicit global $2$-homogeneous solutions to the $p$-Laplacian obstacle problem whose free boundaries have a corner at the origin. In particular, we show that the free boundary is in general not $C^1$ at points where $ abla varphi=0$. On the other hand, under the concavity assumption $| abla varphi|^{2-p}Delta_p varphi<0$, we show the free boundary is countably $(n-1)$-rectifiable and we prove a nondegeneracy property for $u$ at all free boundary points.
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 study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-Delta)^s u=f$ in $Omega$, $mathcal N_s u=0$ in $Omega^c$, then $u$ is $C^alpha$ up tp the boundary for some $alpha>0$. Moreover, in case $s>frac12$, we then show that $uin C^{2s-1+alpha}(overlineOmega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.
In this paper we give a full classification of global solutions of the obstacle problem for the fractional Laplacian (including the thin obstacle problem) with compact coincidence set and at most polynomial growth in dimension $N geq 3$. We do this in terms of a bijection onto a set of polynomials describing the asymptotics of the solution. Furthermore we prove that coincidence sets of global solutions that are compact are also convex if the solution has at most quadratic growth.
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