arXiv subjects

Antonio Farah

Publications and source records attributed to Antonio Farah.

3 recordsLinked to original sources

arXiv2026

A Degenerate One-Phase Free Boundary Problem Arising From the Alt-Phillips Equation for Negative Powers

We study viscosity solutions for a class of degenerate one-phase free boundary problems of the form $wΔw = h(\nabla w)$. We assume the existence of a star-shaped domain $D$ such that $h < 0$ in $D$, $h = 0$ on $\partial D$, and $h > 0$ in $\bar{D}^{c}$. This class of degenerate one-phase free boundary problems arises when a canonical transformation is performed to a semilinear equation $Δu = f(u)$, and $f$ behaves like $-γu^{-(γ+ 1)}$ for some $γ\in (0,2)$. In this case, known as the Alt-Phillips equation for negative power potentials, $h(ρ) = c(|ρ|^2 - 1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $γ$ converges to $2$, the free boundary converges to a minimal surface.

math.AP↗

arXiv2025

On the Grad-Mercier equation and Semilinear Free Boundary Problems

In this paper, we establish regularity and uniqueness results for Grad-Mercier type equations that arise in the context of plasma physics. We show that solutions of this problem naturally develop a dead core, which corresponds to the set where the solutions become identically equal to their maximum. We prove uniqueness, sharp regularity, and non-degeneracy bounds for solutions under suitable assumptions on the reaction term. Of independent interest, our methods allow us to prove that the free boundaries of a broad class of semilinear equations have locally finite $H^{n-1}$ measure.

math.AP↗

arXiv2023

Quantifying and managing uncertainty in piecewise-deterministic Markov processes

In piecewise-deterministic Markov processes (PDMPs) the state of a finite-dimensional system evolves continuously, but the evolutive equation may change randomly as a result of discrete switches. A running cost is integrated along the corresponding piecewise-deterministic trajectory up to the termination to produce the cumulative cost of the process. We address three natural questions related to uncertainty in cumulative cost of PDMP models: (1) how to compute the Cumulative Distribution Function (CDF) of the cumulative cost when the switching rates are fully known; (2) how to accurately bound the CDF when the switching rates are uncertain; and (3) assuming the PDMP is controlled, how to select a control to optimize that CDF. In all three cases, our approach requires posing a system of suitable hyperbolic partial differential equations, which are then solved numerically on an augmented state space. We illustrate our method using simple examples of trajectory planning under uncertainty for several 1D and 2D first-exit time problems. In the Appendix, we also apply this method to a model of fish harvesting in an environment with random switches in carrying capacity.

math.OC↗