arXiv subjects

Farhan Abedin

Publications and source records attributed to Farhan Abedin.

11 recordsLinked to original sources

arXiv2026

Variational properties of Perron's extremal solutions in the Bernoulli one-phase problem

We study the Perron extremal solutions of the stationary one-phase Bernoulli problem. These solutions are important in several applications, but not much is known about the regularity of their free boundaries due to their non-variational construction. We show that, in fact, Perron extremal solutions retain some variational features; specifically, they are solutions in the sense of inner variations. We establish this property by showing that extremal solutions arise as infinite-time limits of monotone solutions of the parabolic Bernoulli problem. As a consequence, we are able to describe the structure of singularities for Perron extremal solutions in two dimensions: largest subsolutions have smooth free boundaries, while smallest supersolutions have free boundary points that are either smooth, or whose blow-up limits are two-plane wedges with equal slope.

math.AP↗

arXiv2025

Boundary regularity for subelliptic equations in the Heisenberg group

We prove boundary Hölder and Lipschitz regularity for a class of degenerate elliptic, second order, inhomogeneous equations in non-divergence form structured on the left-invariant vector fields of the Heisenberg group. Our focus is on the case of operators with bounded and measurable coefficients and bounded right-hand side; when necessary, we impose a dimensional restriction on the ellipticity ratio and a growth rate for the source term near characteristic points of the boundary. For solutions in the characteristic half-space $\{t>0\}$, we obtain an intrinsic second order expansion near the origin when the source term belongs to an appropriate weighted $L^{\infty}$ space; this is a new result even for the frequently studied sub-Laplacian.

math.AP↗

arXiv2025

Regularity of two-phase free boundary minimizers in periodic media

We study the regularity of minimizers of a two-phase energy functional in periodic media. Our main result is a large scale Lipschitz estimate. We also establish improvement-of-flatness for non-degenerate minimizers, which is a key ingredient in the proof of the Lipschitz estimate. As a consequence, we obtain a Liouville property for entire non-degenerate minimizers.

math.AP↗

arXiv2022

Quantitative convergence of the "bulk'' free boundary in an oscillatory obstacle problem

We consider an oscillatory obstacle problem where the coincidence set and free boundary are also highly oscillatory. We establish a rate of convergence for a regularized notion of free boundary to the free boundary of a corresponding classical obstacle problem, assuming the latter is regular. The convergence rate is linear in the minimal length scale determined by the fine properties of a corrector function.

math.AP↗

arXiv2021

A perturbative approach to the parabolic optimal transport problem for non-MTW costs

Fix a pair of smooth source and target densities $ρ$ and $ρ^*$ of equal mass, supported on bounded domains $Ω, Ω^* \subset \mathbb{R}^n$. Also fix a cost function $c_0 \in C^{4,α}(\overlineΩ \times \overline{Ω^*})$ satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume $Ω$ and $Ω^*$ are uniformly $c_0$- and $c_0^*$-convex with respect to each other. We consider a parabolic version of the optimal transport problem between $(Ω,ρ)$ and $(Ω^*,ρ^*)$ when the cost function $c$ is a sufficiently small $C^4$ perturbation of $c_0$, and where the size of the perturbation depends on the given data. Our main result establishes global-in-time existence of a solution $u \in C^2_xC^1_t(\overlineΩ\times [0, \infty))$ of this parabolic problem, and convergence of $u(\cdot,t)$ as $t \to \infty$ to a Kantorovich potential for the optimal transport map between $(Ω,ρ)$ and $(Ω^*,ρ^*)$ with cost function $c$. A noteworthy aspect of our work is that $c$ does \emph{not} necessarily satisfy the weak Ma-Trudinger-Wang condition.

math.AP↗

arXiv2020

Regularity for a special case of two-phase Hele-Shaw flow via parabolic integro-differential equations

We establish that the $C^{1,γ}$ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of regularity mechanism for solutions to a special case of a two-phase free boundary flow related to Hele-Shaw. The special case is due to both a graph assumption on the free boundary of the flow and an assumption that the free boundary is $C^{1,\text{Dini}}$ in space. The free boundary then must immediately become $C^{1,γ}$ for a universal $γ$ depending upon the Dini modulus of the gradient of the graph. These results also apply to one-phase problems of the same type.

math.AP↗

arXiv2020

Inverse Iteration for the Monge-Ampère Eigenvalue Problem

We present an iterative method based on repeatedly inverting the Monge-Ampère operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain $Ω\subset \mathbb{R}^n$. We prove that the iterates $u_k$ generated by this method converge as $k \to \infty$ to a solution of the Monge-Ampère eigenvalue problem $$\begin{cases} \text{det} D^2u = λ_{MA} (-u)^n & \quad \text{in } Ω,\\ u = 0 & \quad \text{on } \partial Ω. \end{cases}$$ Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates $\hat{u}_k := \frac{u_k}{||u_k||_{L^{\infty}(Ω)}}$ converge to the eigenfunction of unit height. In addition, we show that $\lim\limits_{k \to \infty} R(u_k) = \lim\limits_{k \to \infty} R(\hat{u}_k) = λ_{MA}$, where the Rayleigh quotient $R(u)$ is defined as $$R(u) := \frac{\int_Ω (-u) \ \text{det} D^2u}{\int_Ω (-u)^{n+1}}.$$ Our method converges for a wide class of initial choices $u_0$ that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Ampère eigenvalue $λ_{MA}$.

math.AP↗

arXiv2019

Harnack inequality for a class of Kolmogorov-Fokker-Planck equations in non-divergence form

We prove invariant Harnack inequalities for certain classes of non-divergence form equations of Kolmogorov type. The operators we consider exhibit invariance properties with respect to a homogeneous Lie group structure. The coefficient matrix is assumed either to satisfy a Cordes-Landis condition on the eigenvalues, or to admit a uniform modulus of continuity.

math.AP↗

arXiv2018

Exponential Convergence of Parabolic Optimal Transport on Bounded Domains

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We derive a differential Harnack inequality for a special class of functions that solve the linearized problem. Using this Harnack inequality and certain techniques specific to mass transport, we control the oscillation in time of solutions to the parabolic equation, and obtain exponential convergence. Additionally, in the course of the proof, we present a connection with the pseudo-Riemannian framework introduced by Kim and McCann in the context of optimal transport, which is interesting in its own right.

math.AP↗

arXiv2017

Harnack's inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack's inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach from [DGL08] to obtain Harnack's inequality.

math.AP↗

arXiv2016

$C^{1,α}$ estimates for the parallel refractor

We consider the parallel refractor problem when the planar radiating source lies in a medium having higher refractive index than the medium in which the target is located. We prove local $C^{1,α}$ estimates for parallel refractors under suitable geometric assumptions on the source and target, and under local regularity hypotheses on the target set. We also discuss existence of refractors under energy conservation assumptions.

math.AP↗