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Register a new userObstacle problems and free boundaries: an overview
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Authors
Xavier Ros-Oton
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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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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 $nleq4$.
We study the obstacle problem for parabolic operators of the type $partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-Delta)^s$, in the supercritical regime $s in (0,frac{1}{2})$. The best result in this context was due to Caffarelli and Figalli, who established the $C^{1,s}_x$ regularity of solutions for the case $L = (-Delta)^s$, the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually textit{more} regular than in the elliptic case. More precisely, we show that they are $C^{1,1}$ in space and time, and that this is optimal. We also deduce the $C^{1,alpha}$ regularity of the free boundary. Moreover, at all free boundary points $(x_0,t_0)$, we establish the following expansion: $$(u - varphi)(x_0+x,t_0+t) = c_0(t - acdot x)_+^2 + O(t^{2+alpha}+|x|^{2+alpha}),$$ with $c_0 > 0$, $alpha > 0$ and $a in mathbb R^n$.
We study homogenization of a boundary obstacle problem on $ C^{1,alpha} $ domain $D$ for some elliptic equations with uniformly elliptic coefficient matrices $gamma$. For any $ epsiloninmathbb{R}_+$, $partial D=Gamma cup Sigma$, $Gamma cap Sigma=emptyset $ and $ S_{epsilon}subset Sigma $ with suitable assumptions, we prove that as $epsilon$ tends to zero, the energy minimizer $ u^{epsilon} $ of $ int_{D} |gamma abla u|^{2} dx $, subject to $ ugeq varphi $ on $ S_{varepsilon} $, up to a subsequence, converges weakly in $ H^{1}(D) $ to $ widetilde{u} $ which minimizes the energy functional $int_{D}|gamma abla u|^{2}+int_{Sigma} (u-varphi)^{2}_{-}mu(x) dS_{x}$, where $mu(x)$ depends on the structure of $S_{epsilon}$ and $ varphi $ is any given function in $C^{infty}(overline{D})$.
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 $a>0$.
In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,alpha}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g in C^alpha$, and $partialOmega, partialOmega^*in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Amp`ere equations cite{CLW1}, our second result shows that the free boundaries are $C^{2,alpha}$. As an application, in the last we also obtain these regularities of the free boundary in an optimal transport problem with two separate targets.
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