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Gonzalo Dávila

Publications and source records attributed to Gonzalo Dávila.

8 recordsLinked to original sources
arXiv2023

Large harmonic functions for fully nonlinear fractional operators

We study existence, uniqueness and boundary blow-up profile for fractional harmonic functions on a bounded smooth domain $Ω\subset \mathbb R^N$. We deal with harmonic functions associated to uniformly elliptic, fully nonlinear nonlocal operators, including the linear case $$ (-Δ)^s u = 0 \quad \mbox{in} \ Ω, $$ where $(-Δ)^s$ denotes the fractional Laplacian of order $2s \in (0,2)$. We use the viscosity solution's theory and Perron's method to construct harmonic functions with zero exterior condition in $\bar Ω^c$, and boundary blow-up profile $$ \lim_{x\to x_0, x \in Ω}\mathrm{dist}(x, \partial Ω)^{1-s}u(x)=h(x_0), \quad \mbox{for all} \quad x_0\in \partial Ω, $$ for any given boundary data $h \in C(\partial Ω)$. Our method allows us to provide blow-up rate for the solution and its gradient estimates. Results are new even in the linear case.

math.AP↗
arXiv2022

Periodic homogenization of the principal eigenvalue of second-order elliptic operators

In this paper we investigate homogenization results for the principal eigenvalue problem associated to $1$-homogeneous, uniformly elliptic, second-order operators. Under rather general assumptions, we prove that the principal eigenpair associated to an oscillatory operator converges to the eigenpair associated to the effective one. This includes the case of fully nonlinear operators. Rates of convergence for the eigenvalues are provided for linear and nonlinear problems, under extra regularity/convexity assumptions. Finally, a linear rate of convergence (in terms of the oscillation parameter) of suitably normalized eigenfunctions is obtained for linear problems.

math.AP↗
arXiv2022

On large solutions for fractional Hamilton-Jacobi equations

We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated to fully nonlinear elliptic equations of order $2s$, with $s\in (1/2,1)$, and a coercive gradient term with subcritical power $0<p<2s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0<p<2s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry-Lions for the case subquadratic case $1<p<2$.

math.AP↗
arXiv2020

The Nonlocal Inverse Problem of Donsker and Varadhan

In this paper we prove a nonlocal version of the celebrated Inverse Problem of Donsker and Varadhan~\cite{DV} for nonlocal elliptic operators of the form $$ \cL u = L_K u + \cB_K (h,u), $$ where $L_K$ is a uniformly elliptic nonlocal operator with smooth coefficients, and, for $h:\R^N \to \R$ smooth and bounded, $\cB_K(h,u)$ is the associated bilinear form, which can be regarded as a nonlocal transport term.

math.AP↗
arXiv2020

Regularity for parabolic systems with critical growth in the gradient and applications

Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove Hölder regularity of weak solutions.

math.AP↗
arXiv2012

Regularity for solutions of non local parabolic equations

We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We proof Holder regularity in space and time and for translation invariant equations and under different assumptions on the kernels Holder regularity for the spatial derivatives. The proofs rely on a weak parabolic ABP inspired in recent work done by L. Silvestre and the classic ideas of K. Tso and L. Wang. Our results remain uniform as the order of the equation goes to 2 allowing us to understand the non local theory as an extension to the classical one.

math.AP↗