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Héctor A. Chang-Lara

Publications and source records attributed to Héctor A. Chang-Lara.

8 recordsLinked to original sources
arXiv2025

Fractional $Q$-curvature on the sphere and optimal partitions

We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional $Q$-curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new Hölder regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly-coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least-energy solutions as the coupling parameters approach negative infinity.

math.AP↗
arXiv2022

Classical solutions to integral equations with zero order kernels

We show global and interior higher-order log-Hölder regularity estimates for solutions of Dirichlet integral equations where the operator has a nonintegrable kernel with a singularity at the origin that is weaker than that of any fractional Laplacian. As a consequence, under mild regularity assumptions on the right hand side, we show the existence of classical solutions of Dirichlet problems involving the logarithmic Laplacian and the logarithmic Schrödinger operator.

math.AP↗
arXiv2022

Hölder regularity for non-variational porous media type equations

We present a Krylov-Safonov theory approach for the Hölder regularity of viscosity solutions to non-variational porous media type equations. We explore the peculiarity of this type of problem: either the equation falls in a uniformly elliptic regime or the eikonal mechanism takes care of the regularity. Our techniques are based on sliding paraboloids resulting in an ABP-type measure estimate. By combining such estimates, a diminishing of oscillation property is available, resulting in a regularity control in Hölder spaces.

math.AP↗
arXiv2020

Non-convex Hamilton-Jacobi equations with gradient constraints

We study non-convex Hamilton-Jacobi equations in the presence of gradient constraints and produce new, optimal, regularity results for the solutions. A distinctive feature of those equations regards the existence of a lower bound to the norm of the gradient; it competes with the elliptic operator governing the problem, affecting the regularity of the solutions. This class of models relates to various important questions and finds applications in several areas; of particular interest is the modeling of optimal dividends problems for multiple insurance companies in risk theory and singular stochastic control in reversible investment models.

math.AP↗
arXiv2016

Shape Theorems for Poisson Hail on a Bivariate Ground

We consider the extension of the Euclidean stochastic geometry Poisson Hail model to the case where the service speed is zero in some subset of the Euclidean space and infinity in the complement. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process and on the geometrical properties of the zero speed set.

math.PR↗
arXiv2016

From the free boundary condition for Hele-Shaw to a fractional parabolic equation

We propose a method to determine the smoothness of sufficiently flat solutions of one phase Hele-Shaw problems. The novelty is the observation that under a flatness assumption the free boundary --represented by the hodograph transform of the solution- solves a nonlinear integro-differential equation. This nonlinear equation is linearized to a (nonlocal) parabolic equation with bounded measurable coefficients, for which regularity estimates are available. This fact is used to prove a regularity result for the free boundary of a weak solution near points where the solution looks sufficiently flat. More concretely, flat means that in a parabolic neighborhood of the point the solution lies between the solutions corresponding to two parallel flat fronts a small distance apart --a condition that only depends on the the local behavior of the solution. In a neighborhood of such a point, the free boundary is given by the graph of a function whose spatial gradient enjoys a universal Hölder estimate in both space and time.

math.AP↗