arXiv subjects

Anthony Salib

Publications and source records attributed to Anthony Salib.

4 recordsLinked to original sources

arXiv2026

ACF Almost Monotonicity at Infinity with Applications to Perturbed Global Solutions

We study the large-scale behavior of the coincidence set of perturbations of global solutions to the classical obstacle problem in $\mathbb{R}^n\setminus B_1$, with blow-down invariant in the $e_n$ direction. In dimensions $n\geq 3$, we prove that, locally around regular points sufficiently far out, the cross-sections of $\{u=0\}$ perpendicular to $e_n$ are $C^2$ perturbations of ellipsoids. The main ingredient is a new large-scale almost monotonicity formula for the Alt--Caffarelli--Friedman functional. In contrast with the classical small-scale perturbative theory, our argument exploits the stability of the obstacle problem together with the fact that local perturbations vanish under blow-down. The method provides a model mechanism for controlling errors at infinity in stable free boundary problems.

math.AP↗

arXiv2024

A singular perturbation approach to the Dirichlet-area minimisation problem

We study both one and two-phase minimisers of the Dirichlet-area energy $$E(v) = \int_{B_1} \vert\nabla v\vert^2 + Per(\{v>0\},B_1).$$ In the two-phase case, we show that the energies $$E_{\varepsilon}(v) = \int_{B_1}\vert\nabla v\vert^2 + \frac{1}{\varepsilon}W\left(\frac{v}{\varepsilon^{1/2}}\right),$$ $\Gamma$-converge to $E$ as $\varepsilon \to 0$, where $W$ is the double well potential extended by zero outside of $[-1,1]$ . As a consequence, we show that bounded local minimisers of $E_{\varepsilon}$ converge to a local minimiser of $E$.

math.AP↗

arXiv2024

Classification of global solutions to the obstacle problem in the plane

Global solutions to the obstacle problem were first completely classified in two dimensions by Sakai using complex analysis techniques. Although the complex analysis approach produced a very succinct proof in two dimensions, it left the higher dimensional cases, and even closely related problems in two dimensions, unresolved. A complete classification in dimensions $n\geq 3$ was recently given by Eberle, Figalli and Weiss, forty years after Sakai published his proof. In this paper we give a proof of Sakai's classification result for unbounded coincidence sets in the spirit of the recent proof by Eberle, Figalli and Weiss. Our approach, in particular, avoids the need for complex analysis techniques and offers new perspectives on two-dimensional problems that complex analysis cannot address.

math.AP↗

arXiv2024

On the viscosity linearization method without compactness

Savin's small perturbation approach has had far reaching applications in the theory of non-linear elliptic and parabolic PDE. In this short note, we revisit his seminal proof of De-Giorgi's improvement of flatness theorem for minimal surfaces and provide an approach based on the Harnack inequality that avoids the use of compactness arguments.

math.AP↗