arXiv subjects
Publications and source records attributed to Alessio Figalli.
We prove that, for general cost functions on $\mathbb{R}^n$, or for the cost $d^2/2$ on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.
We prove that $C^{1,α}$ $s$-minimal surfaces are automatically $C^\infty$. For this, we develop a new bootstrap regularity theory for solutions of integro-differential equations of very general type, which we believe is of independent interest.
In this paper we present a survey concerning unconstrained free boundary problems of type $$ \left\{ \begin{array}{ll} F_1(D^2u,\nabla u,u,x)=0 & \text{in }B_1 \cap Ω,\\ F_2 (D^2 u,\nabla u,u,x)=0 & \text{in }B_1\setminusΩ,\\ u \in \mathbb{S}(B_1), \end{array} \right. $$ where $B_1$ is the unit ball, $Ω$ is an unknown open set, $F_1, F_2$ are elliptic operators (admitting regular solutions), and $\mathbb{S}$ is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to the optimal regularity of solutions to the above matching problems, and list several open problems in this direction.
This text contains the material discussed by the author in the Bourbaki seminar of June 2018, on the recent developments in the theory of the Monge-Ampère equation.
To generalize the notion of distribution function to dimension $d\geq 2$, in the recent papers it was proposed a concept of center-outward distribution function based on optimal transportation ideas, and the inferential properties of the corresponding center-outward quantile function were studied. A crucial tool needed to derive the desired inferential properties is the continuity and invertibility for the center-outward quantile function outside the origin, as this ensures the existence of closed and nested quantile contours. The aim of this paper is to prove such a continuity and invertibility result.
We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal{L} u= f(u)$, posed in a bounded domain $Ω\subset {\mathbb R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal{L}$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $(-Δ)^s$ ($0<s<1$) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity $f$ is increasing and looks like a power function $f(u)\sim u^p$, with $p\le 1$. The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number $\frac{2s}{1-p}$ goes below the exponent $γ\in(0,1]$ corresponding to the Hölder regularity of the first eigenfunction $\mathcal{L}Φ_1=λ_1 Φ_1$. Indeed a change of boundary regularity happens in the different regimes $\frac{2s}{1-p} \gtreqqless γ$, and in particular a logarithmic correction appears in the "critical" case $\frac{2s}{1-p} = γ$. Indeed a change of boundary regularity happens in the different regimes $\frac{2s}{1-p} \gtreqqless γ$, and in particular a logarithmic correction appears in the "critical" case $\frac{2s}{1-p} = γ$. For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range $0<s\leq \frac{1-p}{2}$.
In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers \cite{C77,C98}, regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of $C^1$ manifolds of varying dimension. In two dimensions, this $C^1$ result has been improved to $C^{1,α}$ by Weiss \cite{W99}. In this paper we prove that, for $n=2$ singular points are locally contained in a $C^2$ curve. In higher dimension $n\ge 3$, we show that the same result holds with $C^{1,1}$ manifolds (or with countably many $C^2$ manifolds), up to the presence of some "anomalous" points of higher codimension. In addition, we prove that the higher dimensional stratum is always contained in a $C^{1,α}$ manifold, thus extending to every dimension the result in \cite{W99}. We note that, in terms of density decay estimates for the contact set, our result is optimal. In addition, for $n\ge3$ we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.
This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + {\mathcal L}u^m=0$, $m>1$, where the operator ${\mathcal L}$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $Ω\subset{\mathbb R}^N$. As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, ${\mathcal L}$ can be a power of a uniformly elliptic operator with $C^1$ coefficients. Since the nonlinearity is given by $u^m$ with $m>1$, the equation is degenerate parabolic. The basic well-posedness theory for this class of equations has been recently developed in [14,15]. Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when ${\mathcal L}$ is a uniformly elliptic operator, and provide new estimates even in this setting. A surprising aspect discovered in this paper is the possible presence of non-matching powers for the long-time boundary behavior. More precisely, when ${\mathcal L}=(-Δ)^s$ is a spectral power of the {Dirichlet} Laplacian inside a smooth domain, we can prove that: - when $2s> 1-1/m$, for large times all solutions behave as ${\rm dist}^{1/m}$ near the boundary; - when $2s\le 1-1/m$, different solutions may exhibit different boundary behavior. This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation ${\mathcal L}u^m=u$.
A developable cone ("d-cone") is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance $ε$. Starting from a nonlinear model depending on the thickness $h > 0$ of the sheet, we prove a $Γ$-convergence result as $h \rightarrow 0$ to a fourth-order obstacle problem for curves in $\mathbb{S}^2$. We then describe the exact shape of minimizers of the limit problem when $ε$ is small. In particular, we rigorously justify previous results in the physics literature.
We prove that every bounded stable solution of \[ (-Δ)^{1/2} u + f(u) =0 \qquad \mbox{in }\mathbb R^3\] is a 1D profile, i.e., $u(x)= ϕ(e\cdot x)$ for some $e\in \mathbb S^2$, where $ϕ:\mathbb R\to \mathbb R$ is a nondecreasing bounded stable solution in dimension one. This proves the De Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we give a positive answer to the De Giorgi conjecture for boundary reactions in $\mathbb R^{d+1}_+=\mathbb R^{d+1}\cap \{x_{d+1}\geq 0\}$ when $d = 4$, by proving that all critical points of $$ \int_{\{x_{d+1\geq 0}\}} \frac12 |\nabla U|^2 \,dx\, dx_{d+1} + \int_{\{x_{d+1}=0\}} \frac 1 4 (1-U^2)^2 \,dx $$ that are monotone in $\mathbb R^d$ (that is, up to a rotation, $\partial_{x_d} U>0$) are one dimensional. Our result is analogue to the fact that stable embedded minimal surfaces in $\mathbb R^3$ are planes. Note that the corresponding result about stable solutions to the classical Allen-Cahn equation (namely, when the half-Laplacian is replaced by the classical Laplacian) is still open.
We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $φ$ satisfies $Δφ\leq 0$ near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a $(n-1)$-dimensional $C^{1,α}$ manifold by the results in \cite{CSS}, and a set of singular points, which we prove to be contained in a union of $k$-dimensional $C^1$-submanifold, $k=0,\ldots,n-1$. Such a complete result on the structure of the free boundary was known only in the case of the classical Laplacian \cite{C-obst1,C-obst2}, and it is new even for the Signorini problem (which corresponds to the particular case of the $\frac12$-fractional Laplacian). A key ingredient behind our results is the validity of a new non-degeneracy condition $\sup_{B_r(x_0)}(u-φ)\geq c\,r^2$, valid at all free boundary points $x_0$.
We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $\min\bigl\{-Δ_p u,\,u-φ\bigr\}=0$ in $Ω\subset\mathbb R^n$. Here, $Δ_p u=\textrm{div}\bigl(|\nabla u|^{p-2}\nabla u\bigr)$, and $p\in(1,2)\cup(2,\infty)$. Near those free boundary points where $\nabla φ\neq0$, the operator $Δ_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\nabla φ=0$ then $Δ_p$ 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 $\nabla φ=0$. On the one hand, for every $p\neq2$ 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 $\nabla φ=0$. On the other hand, under the "concavity" assumption $|\nabla φ|^{2-p}Δ_p φ<0$, we show the free boundary is countably $(n-1)$-rectifiable and we prove a nondegeneracy property for $u$ at all free boundary points.
We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble. As an application of our result, we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in $\mathbf{R}^n$ related to the Yamabe flow.
We prove a strong form of the quantitative Sobolev inequality in $\mathbb{R}^n$ for $p\geq 2$, where the deficit of a function $u\in \dot W^{1,p} $ controls $\| \nabla u -\nabla v\|_{L^p}$ for an extremal function $v$ in the Sobolev inequality.
The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square of the volume of the symmetric difference between the set and a ball. This sharp result is strictly related to the physically significant problem of understanding near maximizers in the Riesz rearrangement inequality with a strictly decreasing radially decreasing kernel. Moreover, it implies as a particular case the sharp quantitative Euclidean isoperimetric inequality from \cite{fuscomaggipratelli}.
We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-Δ)^su^m=0$ in $(0,\infty)\timesΩ$, for $m>1$ and $s\in (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,\infty)\times({\mathbb R}^N\setminusΩ)$, and nonnegative initial condition $u(0,\cdot)=u_0\geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $\partialΩ$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^\infty$ in $x$ and $C^{1,α}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-\mathcal L F(u)=0$ in $Ω$, both in bounded or unbounded domains.
We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \geq 1$ outside of a closed singular set of measure zero. We also establish global $W^{2,p}$ estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost.
In this note we establish some rigidity and stability results for Caffarelli's log-concave perturbation theorem. As an application we show that if a 1-log-concave measure has almost the same Poincaré constant as the Gaussian measure, then it almost splits off a Gaussian factor.