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Luis Caffarelli

Publications and source records attributed to Luis Caffarelli.

At least 37 records · Page 2Linked to original sources
arXiv2016

A logistic equation with nonlocal interactions

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, and transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexistence of solutions in terms of the spectral properties of the domain. In particular, we give a detailed description of the fact that nonlocal populations may better adapt to sparse resources and small environments.

math.AP↗
arXiv2015

On a fractional Monge-Ampère operator

In this paper we consider a fractional analogue of the Monge-Ampère operator. Our operator is a concave envelope of fractional linear operators of the form $ \inf_{A\in \mathcal{A}}L_Au, $ where the set of operators corresponds to all affine transformations of determinant one of a given multiple of the fractional Laplacian. We set up a relatively simple framework of global solutions prescribing data at infinity and global barriers. In our key estimate, we show that the operator remains strictly elliptic, which allows to apply known regularity results for uniformly elliptic operators and deduce that solutions are classical.

math.AP↗
arXiv2015

On a Class of Stochastic Differential Equations With Jumps and Its Properties

We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for a class of these operators. We also establish key connections between the recurrence properties of these jump processes and the non-local partial differential operator. One of the key results is the regularity of solutions of the Dirichlet problem for a class of operators with locally weakly Hölder continuous kernels.

math.PR↗
arXiv2015

Porous Medium Flow with both a Fractional Potential Pressure and Fractional Time Derivative

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account "memory''. The precise model is \[ D_t^α u - \text{div}(u(-Δ)^{-σ} u) = f, \quad 0<σ<1/2. \] We pose the problem over $\{t\in {\mathbb R}^+, x\in {\mathbb R}^n\}$ with nonnegative initial data $u(0,x)\geq 0 $ as well as right hand side $f\geq 0$. We first prove existence for weak solutions when $f,u(0,x)$ have exponential decay at infinity. Our main result is Hölder continuity for such weak solutions.

math.AP↗
arXiv2015

A non local Monge-Ampere equation

We introduce a non local analog to the Monge-Ampere operator and show some of its properties. We prove that a global problem involving this operator has C 1,1 solutions in the full space.

math.AP↗
arXiv2015

A Parabolic Problem with a Fractional-Time Derivative

We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Holder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.

math.AP↗
arXiv2014

Regularity of solutions of the fractional porous medium flow with exponent 1/2

We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla\cdot(u\nabla (-Δ)^{-1/2}u).$ For definiteness, the problem is posed in $\{x\in\mathbb{R}^N, t\in \mathbb{R}\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. Previous papers have established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation, as well as the boundedness of nonnegative solutions with $L^1$ data, for the more general family of equations $u_t=\nabla\cdot(u\nabla (-Δ)^{-s}u)$, $0<s<1$. Here we establish the $C^α$ regularity of such weak solutions in the difficult fractional exponent case $s=1/2$. For the other fractional exponents $s\in (0,1)$ this Hölder regularity has been proved in $[5]$. The method combines delicate De Giorgi type estimates with iterated geometric corrections that are needed to avoid the divergence of some essential energy integrals due to fractional long-range effects.

math.AP↗
arXiv2013

On the asymptotic behavior of a Boltzmann-type price formation model

In this paper we study the asymptotic behavior of a Boltzmann type price formation model, which describes the trading dynamics in a financial market. In many of these markets trading happens at high frequencies and low transactions costs. This observation motivates the study of the limit as the number of transactions $k$ tends to infinity, the transaction cost $a$ to zero and $ka=const$. Furthermore we illustrate the price dynamics with numerical simulations.

math.AP↗
arXiv2013

Minimization of a fractional perimeter-Dirichlet integral functional

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely $$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big),$$ with $σ\in(0,1)$. We obtain regularity results for the minimizers and for their free boundaries $\p \{u>0\}$ using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems.

math.AP↗
arXiv2013

On a Boltzmann type price formation model

In this paper we present a Boltzmann type price formation model, which is motivated by a parabolic free boundary model for the evolution of the prize presented by Lasry and Lions in 2007. We discuss the mathematical analysis of the Boltzmann type model and show that its solutions converge to solutions of the model by Lasry and Lions as the transaction rate tends to infinity. Furthermore we analyse the behaviour of the initial layer on the fast time scale and illustrate the price dynamics with various numerical experiments.

math.AP↗
arXiv2013

Regularity properties of nonlocal minimal surfaces via limiting arguments

We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and $s$ is close to 1.

math.AP↗
arXiv2012

Regularity of solutions of the fractional porous medium flow

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely, $$ u_t=\nabla\cdot(u\nabla (-Δ)^{-s}u), \quad \ 0<s<1. $$ The problem is posed in $\{x\in\ren, t\in \re\}$ with nonnegative initial data $u(x,0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. Here we establish the boundedness and $C^α$ regularity of such weak solutions

math.AP↗
arXiv2011

Non-Local Tug-of-War and the Infinity Fractional Laplacian

Motivated by the "tug-of-war" game studied in [12], we consider a "non-local" version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving of a fixed amount $ε>0$ (as is done in the classical case), it is a $s$-stable Levy process which chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically we derive a deterministic non-local integro-differential equation that we call "infinity fractional Laplacian". We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double obstacle problem, both problems having a natural interpretation as "tug-of-war" games.

math.AP↗
arXiv2011

Regularity of solutions to the parabolic fractional obstacle problem

In this paper we study a parabolic version of the fractional obstacle problem, proving almost optimal regularity for the solution. This problem is motivated by an American option model proposed by Menton which introduces, into the theory of option evaluation, discontinuous paths in the dynamics of the stock's prices.

math.AP↗
arXiv2010

On the Evans-Krylov theorem

In this note, motivated by our work on integral fully nonlinear equations, we provide a variation of the proof of Evans-Krylov theorem about the interior $C^{2,α}$ a priori estimate for concave fully nonlinear elliptic equations. The proof we present is shorter compared to the original proofs, although the key tools used in the argument are the same.

math.AP↗