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Register a new userGeneric regularity of free boundaries for the obstacle problem
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The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $mathbb R^n$. By classical results of Caffarelli, the free boundary is $C^infty$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$-dimensional ---that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero $mathcal H^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). In particular, for $nleq4$, the free boundary is generically a $C^infty$ manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions $nleq4$.
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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 prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.
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.
In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary.
Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, Geometric Measure Theory, and Harmonic Analysis. The most important mathematical challenge in this context is to understand the structure and regularity of free boundaries. In this paper we provide an invitation to this area of research by presenting, in a completely non-technical manner, some classical results as well as some recent results of the author.
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