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Register a new userRegularity of local minimizers of the interaction energy via obstacle problems
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The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems.
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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 $a in mathbb R^n$.
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 investigate minimizers defined on a bounded domain in $mathbb{R}^2$ for the Maier--Saupe Q--tensor energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Mujamdars modification of the Ginzburg--Landau Q--tensor model, so as to constrain the competing states to have eigenvalues in the closure of a physically realistic range. We prove that minimizers are regular and in several model problems we are able to use this regularity to prove that minimizers have eigenvalues strictly within the physical range.
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 $a>0$.
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