arXiv subjects

Begoña Barrios

Publications and source records attributed to Begoña Barrios.

14 recordsLinked to original sources

arXiv2025

Propagation in the Fisher-KPP equation with Mixed Operator

Our investigation focuses on the asymptotic spreading behavior of the Fisher-KPP equation with a mixed local-nonlocal operator in the diffusion (see the work by X. Cabré and J.-M. Roquejoffre, 2013, ref.[8]) to the setting of mixed diffusion, which involves both the classical and the fractional Laplacian in order to analyze the long-time dynamics of the equation. A key step in our approach involves the construction and detailed study of the heat kernel associated with the mixed operator, which we use to develop a theory of mild solutions and establish a comparison principle in suitable weighted function spaces. This framework allows us to rigorously establish the non-existence of traveling waves and characterize the large-time spreading rate of solutions. We show that the influence of the fractional Laplacian dominates over the classical Laplacian, especially in the initial layer, where it dictates the exponential propagation rate and the thickness of the solution tails.

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arXiv2025

Higher regularity in nonlocal free boundary problems

We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,α}$ free boundaries are $C^\infty$. This is new even for the fractional Laplacian, as it was only known in case $s=\frac12$. We also establish a general result for overdetermined problems, showing that if the boundary condition is smooth, then so is $\partialΩ$. Our approach is very robust and works as well for the nonlocal obstacle problem, where it yields a new proof of the higher regularity of free boundaries, completely different from the one in [AbRo20]. In order to prove our results, we need to develop, among other tools, new integration by parts formulas and delicate boundary Hölder estimates for nonlocal equations with (local) Neumann boundary conditions that had not been studied before and are of independent interest.

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arXiv2025

Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation

In this paper, we investigate a mixed elliptic equation involving both local and nonlocal Laplacian operators, with a power-type nonlinearity. Specifically, we consider a Lane-Emden type equation of the form \[-Δu + (-Δ)^s u = u^p,\quad\mbox{ in }\mathbb{R}^n.\] where the operator combines the classical Laplacian and the fractional Laplacian. We establish the existence of solutions for exponents slightly below the critical local Sobolev exponent, that is, for $p < \frac{n+2}{n-2}$, with $p$ close to $\frac{n+2}{n-2}$. Our results show that, due to the interaction between the local and nonlocal operators, this mixed Lane-Emden-Fowler equation does not admit a critical exponent in the traditional sense. The existence proof is carried out using a Lyapunov-Schmidt type reduction method and, as far as we know, provide the first example of an elliptic operator for which the duality between critical exponents fails.

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arXiv2024

The limit as $s\nearrow 1$ of the fractional convex envelope

We study the behavior of the fractional convexity when the fractional parameter goes to 1. For any notion of convexity, the convex envelope of a datum prescribed on the boundary of a domain is defined as the largest possible convex function inside the domain that is below the datum on the boundary. Here we prove that the fractional convex envelope inside a strictly convex domain of a continuous and bounded exterior datum converges when $s\nearrow 1$ to the classical convex envelope of the restriction to the boundary of the exterior datum.

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arXiv2024

The evolution problem associated with the fractional first eigenvalue

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as $t\to \infty$ with a bound that is given by the first eigenvalue for this problem that we also study.

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arXiv2022

Periodic fractional Ambrosetti-Prodi for one-dimensional problem with drift

We establish Ambrosetti -Prodi type results for periodic solutions of one -dimensional nonlinear problems with drift term and drift -less whose principal operator is the fractional Laplacian of order $s\in(0,1)$. We establish conditions for the existence and nonexistence of solutions. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also establish a priori bounds in order to get multiplicity results. We also prove that the solutions are $C^{1,α}$ under some regularity assumptions in the nonlinearities, that is, the solutions of equations are classical. We finish the work obtaining existence results for problems with the fractional Laplacian with singular nonlinearity. In particular, we establish an Ambrosetti-Prodi type problem with singular nonlinearities.

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arXiv2020

Equivalence of weak and viscosity solutions in fractional non-homogeneous problems

We establish the equivalence between the notions of weak and viscosity solutions for non-homogeneous equations whose main operator is the fractional p-Laplacian and the lower order term depends on $x$, $u$ and $D_s^p u$, being the last one a type of fractional derivative.

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arXiv2020

Study of the existence of supersolutions for nonlocal equations with gradient terms

We study the existence of positive supersolutions of nonlocal equations $(-Δ)^s u+ |\nabla u|^q=λf(u)$ in exterior domains where the datum $f$ can be comparade with $u^{p}$ near the origin. We prove that the existence or bounded supersolutions its depend of the values of $p$, $q$ and $s$.

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arXiv2017

Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

We present some comparison results for solutions to certain non local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non local version of the results obtained by J. Dávila and J. Dávila-L. Dupaigne for the classical case respectively.

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arXiv2017

Global regularity for the free boundary in the obstacle problem for the fractional Laplacian

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 $x_0$.

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arXiv2016

Free boundary regularity in the parabolic fractional obstacle problem

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump Lévy 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 $C^{1,α}$ graph in $x$ and $t$ near any regular free boundary point $(x_0,t_0)\in \partial\{u>φ\}$. 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 \[u(x,t)-φ(x)=c_0\bigl((x-x_0)\cdot e+a(t-t_0)\bigr)_+^{1+s}+o\bigl(|x-x_0|^{1+s+α}+ |t-t_0|^{1+s+α}\bigr),\] with $c_0>0$, $e\in \mathbb{S}^{n-1}$, and $a>0$.

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arXiv2014

Regularization for the supercritical quasi-geostrophic equation

Motivated by the De Giorgi type argument used in a recent paper by Caffarelli and Vasseur, we prove Hölder-regularity for weak solutions of the supercritical quasi-geostrophic equation with minimal assumptions on the initial datum.

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arXiv2014

On the moving plane method for nonlocal problems in bounded domains

We consider a nonlocal problem involving the fractional laplacian and the Hardy potential, in bounded smooth domains. Exploiting the moving plane method and some weak and strong comparison principles, we deduce symmetry and monotonicity properties of positive solutions under zero Dirichlet boundary conditions.

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arXiv2013

A Widder's type Theorem for the heat equation with nonlocal diffusion

The main goal of this work is to prove that every non-negative {\it strong solution} $u(x,t)$ to the problem $$ u_t+(-Δ)^{α/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<α<2, $$ can be written as $$u(x,t)=\int_{\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\, dy},$$ where $$P_{t}(x)=\frac{1}{t^{n/α}}P\left(\frac{x}{t^{1/α}}\right), $$ and $$ P(x):=\int_{\mathbb{R}^{n}}{e^{ix\cdotξ-|ξ|^α}dξ}. $$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by D. V. Widder in \cite{W0} to the nonlocal diffusion framework.

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