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Luis A. Caffarelli

Publications and source records attributed to Luis A. Caffarelli.

16 recordsLinked to original sources
arXiv2022

Interaction of a free boundary with a diffusion on a plane: analogy with the obstacle problem

A horizontal $N$-dimensional plane, having a diffusion of its own, exchanges with the lower half space. There, a reaction-diffusion process, modelled by a free boundary problem, takes place. We wish to understand whether, and how, the free boundary meets the plane. The origin of this problem is a two-dimensional reaction diffusion model proposed some time ago by the second author, in collaboration with H. Berestycki and L. Rossi, to model how biological invasions can be enhanced by a line of fast diffusion. Some counter-intuitive numerical simulations of this model, due to A.-C. Coulon, have been explained by the first two authors by transforming the model into a free boundary interacting with a line, and a careful study of the free boundary. At this occasion, it was noticed that the free boundary very much like that of the obstacle problem. The goal of the paper is to explain how this analogy with the obstacle problem can be pushed further in higher space dimensions.

math.AP↗
arXiv2022

On a family of fully nonlinear integro-differential operators: From fractional Laplacian to nonlocal Monge-Ampère

We introduce a new family of intermediate operators between the fractional Laplacian and the Caffarelli-Silvestre nonlocal Monge-Ampère that are given by infimums of integro-differential operators. Using rearrangement techniques, we obtain representation formulas and give a connection to optimal transport. Finally, we consider a global Poisson problem, prescribing data at infinity, and prove existence, uniqueness, and $C^{1,1}$-regularity of solutions in the full space.

math.AP↗
arXiv2017

On a long range segregation model

In this work we study the properties of segregation processes modeled by a family of equations $$ L(u_i) (x) = u_i(x)\: F_i (u_1, \ldots, u_K)(x)\qquad i=1,\ldots, K $$ where $F_i (u_1, \ldots, u_K)(x)$ is a non-local factor that takes into consideration the values of the functions $u_j$'s in a full neighborhood of $x.$ We consider as a model problem $$Δu_i^\ep (x) = \frac1{\ep^2} u_i^\ep (x)\sum_{i\neq j} H(u_j^\ep)(x)$$ where $\ep$ is a small parameter and $H(u_j^\ep)(x)$ is for instance $$H(u_j^\ep)(x)= \int_{\mathcal{B}_1 (x)} u_j^\ep (y)\, \text{d}y$$ or $$H(u_j^\ep)(x)= \sup_{y\in \mathcal{B}_1(x)} u_j^\ep (y).$$ Here the set $\mathcal{B}_1(x)$ is the unit ball centered at $x$ with respect to a smooth, uniformly convex norm $ρ$ of $\real^n$. Heuristically, this will force the populations to stay at $ρ$-distance 1, one from each other, as $\ep\to0$.

math.AP↗
arXiv2016

On some pointwise inequalities involving nonlocal operators

The purpose of this paper is three-fold: first, we survey on several known pointwise identities involving fractional operators; second, we propose a unified way to deal with those identities; third, we prove some new pointwise identities in different frameworks in particular geometric and infinite-dimensional ones.

math.AP↗
arXiv2016

A minimization problem with free boundary related to a cooperative system

We study the minimum problem for the functional $\int_Ω\bigl( \vert \nabla \mathbf{u} \vert^{2} + Q^{2}χ_{\{\vert \mathbf{u}\vert>0\}} \bigr)dx$ with the constraint $u_i\geq 0$ for $i=1,\cdots,m$ where $Ω\subset\mathbb{R}^{n}$ is a bounded domain and $\mathbf{u}=(u_1,\cdots,u_m)\in H^{1}(Ω;\mathbb{R}^{m})$. Using an array of technical tools, from geometric analysis for the free boundaries, we reduce the problem to its scalar counterpart and hence conclude similar results as that of scalar problem. This can also be seen as the most novel part of the paper, that possibly can lead to further developments of free boundary regularity for systems.

math.AP↗
arXiv2016

A Free Boundary Problem Related to Thermal Insulation

We study a free boundary problem arising from the theory of thermal insulation. The outstanding feature of this set optimization problem is that the boundary of the set being optimized is not a level surface of a harmonic function, but rather a hypersurface along which a harmonic function satisfies a Robin condition. We show that minimal sets exist, satisfy uniform density estimates, and, under some geometric conditions, have "locally flat" boundaries.

math.AP↗
arXiv2013

Regularity for anisotropic fully nonlinear integro-differential equations

We consider fully nonlinear integro-differential equations governed by kernels that have different homogeneities in different directions. We prove a nonlocal version of the ABP estimate, a Harnack inequality and the interior $C^{1, γ}$ regularity, extending the results of [Caffarelli and Silvestre, CPAM 62, 2009] to the anisotropic case.

math.AP↗
arXiv2011

On a price formation free boundary model by Lasry & Lions: The Neumann problem

We discuss local and global existence and uniqueness for the price formation free boundary model with homogeneous Neumann boundary conditions introduced by Lasry & Lions in 2007. The results are based on a transformation of the problem to the heat equation with nonstandard boundary conditions. The free boundary becomes the zero level set of the solution of the heat equation. The transformation allows us to construct an explicit solution and discuss the behavior of the free boundary. Global existence can be verified under certain conditions on the free boundary and examples of non-existence are given.

math.AP↗
arXiv2011

On a price formation free boundary model by Lasry & Lions

We discuss global existence and asymptotic behaviour of a price formation free boundary model introduced by Lasry & Lions in 2007. Our results are based on a construction which transforms the problem into the heat equation with specially prepared initial datum. The key point is that the free boundary present in the original problem becomes the zero level set of this solution. Using the properties of the heat operator we can show global existence, regularity and asymptotic results of the free boundary.

math.AP↗
arXiv2010

Nonlinear porous medium flow with fractional potential pressure

We study a porous medium equation, with nonlocal diffusion effects given by an inverse fractional Laplacian operator. We pose the problem in n-dimensional space for all t>0 with bounded and compactly supported initial data, and prove existence of a weak and bounded solution that propagates with finite speed, a property that is nor shared by other fractional diffusion models.

math.AP↗
arXiv2010

The Problem of Two Sticks

Let $ l =[l_0,l_1]$ be the directed line segment from $l_0\in {\mathbb R}^n$ to $l_1\in{\mathbb R}^n.$ Suppose $\bar l=[\bar l_0,\bar l_1]$ is a second segment of equal length such that $l, \bar l$ satisfy the "two sticks condition": $\| l_1-\bar l_0\| \ge \| l_1-l_0\|, \| \bar l_1-l_0\| \ge \| \bar l_1-\bar l_0\|.$ Here $\| \cdot\| $ is a norm on ${\mathbb R}^n.$ We explore the manner in which $l_1-\bar l_1$ is then constrained when assumptions are made about "intermediate points" $l_* \in l$, $\bar l_* \in \bar l.$ Roughly speaking, our most subtle result constructs parallel planes separated by a distance comparable to $\| l_* -\bar l_*\| $ such that $l_1-\bar l_1$ must lie between these planes, provided that $\| \cdot\| $ is "geometrically convex" and "balanced", as defined herein. The standard $p$-norms are shown to be geometrically convex and balanced. Other results estimate $\| l_1-\bar l_1 \|$ in a Lipschitz or Hölder manner by $\| l_* -\bar l_* \| $. All these results have implications in the theory of eikonal equations, from which this "problem of two sticks" arose.

math.DG↗
arXiv2008

Convergence of nonlocal threshold dynamics approximations to front propagation

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $α\in (0,2)$ converge to moving fronts. When $α\geqq 1$ the resulting interface moves by weighted mean curvature, while for $α<1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.

math.AP↗
arXiv2007

Random homogenization of an obstacle problem

We study the homogenization of an obstacle problem in a perforated domain. The holes are periodically distributed but have random size and shape. The capacity of the holes is assumed to be stationary ergodic. As in the periodic case, we show that the asymptotic behavior of the solutions is described by an elliptic equation involving an additional term that takes into account the effects of the obstacle.

math.AP↗
arXiv2002

Non linear elliptic theory and the Monge-Ampere equation

The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ampere equation, links in some way the ideas comming from the calculus of variations and those of the theory of fully non linear equations.

math.AP↗
arXiv2000

Regularity of a free boundary with application to the Pompeiu problem

In the unit ball B(0,1), let $u$ and $Ω$ (a domain in $\R$) solve the following overdetermined problem: $$Δu =χ_Ω\quad \hbox{in} B(0,1), \qquad 0 \in \partial Ω, \qquad u=|\nabla u |=0 \quad \hbox{in} B(0,1)\setminus Ω,$$ where $χ_Ω$ denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of $Ω$ does not develop cusp singularities at the origin then we prove $\partial Ω$ is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.

math.AP↗