14 papers
Qualitative-to-quantitative regularity for degenerate equations
Aelson Sobral, Eduardo V. Teixeira, José Miguel Urbano
We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the -Laplace equation, , we prove that local $C^…
From qualitative smoothness to uniform estimates for fully nonlinear elliptic equations
Aelson Sobral, Eduardo V. Teixeira
We establish a general mechanism that turns qualitative smoothness into uniform, quantitative regularity estimates for fully nonlinear elliptic equations. The central conclusion is…
Endpoint Differentiability Moduli for Fully Nonlinear Elliptic Equations
Aelson Sobral, Eduardo V. Teixeira
A classical consequence of Caffarelli's fully nonlinear regularity theory [L. A. Caffarelli, Ann. of Math. (2) 130 (1989), no. 1, 189-213] is that viscosity solutions of uniformly…
The Monge--Ampère equation on graphs
Ahmed Alkhozaae, Julio D. Rossi, Aelson Sobral +1
We introduce a version of the Monge--Ampère equation on finite graphs, motivated by nonlinear graph-based interpolation and semi-supervised learning. The operator is defined as the…
Uniform Lipschitz regularity for two-phase singularly perturbed fully nonlinear elliptic equations
Thialita M. Nascimento, Aelson Sobral, Eduardo V. Teixeira
We study sign-changing viscosity solutions of the singularly perturbed fully nonlinear equation $$ F(D^2u_\varepsilon) = \fracα{\varepsilon} β\left(\frac{u_\varepsilon}{\varepsilon…
Lipschitz regularity for the parabolic obstacle problem
David Jesus, Aelson Sobral, José Miguel Urbano
We study the obstacle problem for the parabolic fractional Laplace equation \[\partial_t u+(-Δ_p)^su = 0\] in the degenerate range . We prove that viscosity soluti…
On a Weiss-type Almost Monotonicity Formula
Aelson Sobral
We establish a Weiss-type almost-monotonicity formula for a broad class of variable-coefficient energy functionals, assuming only minimal regularity of the coefficients. As an appl…
Fractional -caloric functions are Lipschitz
David Jesus, Aelson Sobral, José Miguel Urbano
We study the parabolic fractional Laplace equation in the degenerate range . We show that weak solutions are Lipschitz continuous…
Keller-Osserman and Harnack type results for nonlinear elliptic PDE with unbounded ingredients
Boyan Sirakov, Aelson Sobral
We show that the classical Keller-Osserman theorem on the solvability of the equation is valid when is a general operator in divergence form w…
Borderline regularity in singular free boundary problems
Damião J. Araújo, Aelson Sobral, Eduardo V. Teixeira +1
In this paper, we investigate the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term . When is merely bounded an…
Regularity in diffusion models with gradient activation
Damião Araújo, Aelson Sobral, Eduardo V. Teixeira
We prove sharp regularity estimates for solutions of highly degenerate fully nonlinear elliptic equations. These are free boundary models in which a nonlinear diffusion process dri…
Higher regularity of solutions to fully nonlinear elliptic equations
Thialita M. Nascimento, Ginaldo Sá, Aelson Sobral +1
We establish higher regularity properties of solutions to fully nonlinear elliptic equations at interior critical points. The key novelty of our estimates lies in the fact that the…
On free boundary problems shaped by varying singularities
Damião Araújo, Aelson Sobral, Eduardo V. Teixeira +1
We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power and how it ch…
Cordes-Nirenberg type results for nonlocal equations with deforming kernels
Disson dos Prazeres, Aelson Sobral, José Miguel Urbano
We derive Cordes-Nirenberg type results for nonlocal elliptic integro-differential equations with deforming kernels comparable to sections of a convex solution of a Monge-Ampère eq…