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Edgard A. Pimentel

Publications and source records attributed to Edgard A. Pimentel.

At least 19 recordsLinked to original sources
arXiv2026

Numerical methods for fully nonlinear degenerate diffusions

We propose finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of the schemes. Our focus is on the pure equation and a related free boundary problem of transmission type. The cornerstone of our argument is a regularisation procedure. It decouples the degeneracy term from the elliptic operator driving the diffusion process. In the free boundary setting, a two-parameters regularisation entails new, genuine difficulties. To bypass them, we resort to the intrinsic properties of the regularised problem and a fixed-point type of argument. We present numerical experiments supporting our theoretical results. Our methods are flexible, and our approach can be extended to a broader class of non-variational problems.

math.NA↗
arXiv2025

A numerical scheme for a fully nonlinear free boundary problem

We propose a numerical method to approximate viscosity solutions of fully nonlinear free transmission problems. The method discretises a two-layer regularisation of a PDE, involving a functional and a vanishing parameter. The former is handled via a fixed-point argument. We then prove that the numerical method converges to a one-parameter regularisation of the free boundary problem. Regularity estimates enable us to take the vanishing limit of such a parameter and recover a viscosity solution of the free transmission problem. Our main contribution is the design of a computational strategy, based on fixed-point arguments and approximated problems, to solve fully nonlinear free boundary models. We finish the paper with two numerical examples to validate our method.

math.NA↗
arXiv2024

Fully nonlinear free boundary problems: optimal boundary regularity beyond convexity

We study a general class of elliptic free boundary problems equipped with a Dirichlet boundary condition. Our primary result establishes an optimal $C^{1,1}$-regularity estimate for $L^p$-strong solutions at points where the free and fixed boundaries intersect. A key novelty is that no convexity or concavity assumptions are imposed on the fully nonlinear operator governing the system. Our analysis derives BMO estimates in a universal neighbourhood of the fixed boundary. It relies solely on a differentiability assumption. Once those estimates are available, applying by now standard methods yields the optimal regularity.

math.AP↗
arXiv2024

Boundary regularity for a fully nonlinear free transmission problem

We examine boundary regularity for a fully nonlinear free transmission problem. We argue using approximation methods, comparing the operators driving the problem with a limiting profile. Working natural conditions on the data of the problem, we produce regularity estimates in Sobolev and $C^{1,{\rm Log-Lip}}$-spaces. Our findings extend recent developments in the literature to the free boundary setting.

math.AP↗
arXiv2023

A Hessian-dependent functional with free boundaries and applications to mean-field games

We study a Hessian-dependent functional driven by a fully nonlinear operator. The associated Euler-Lagrange equation is a fully nonlinear mean-field game with free boundaries. Our findings include the existence of solutions to the mean-field game, together with Hölder continuity of the value function and improved integrability of the density. In addition, we prove the reduced free boundary is a set of finite perimeter. To conclude our analysis, we prove a $Γ$-convergence result for the functional.

math.AP↗
arXiv2023

BMO-regularity for a degenerate transmission problem

We examine a transmission problem driven by a degenerate quasilinear operator with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis: the absence of representation formulas for the operator and the degenerate nature of the diffusion process. Our arguments circumvent these difficulties and lead to new regularity estimates. For bounded interface data, we prove the local boundedness of weak solutions and establish an estimate for their gradient in ${\rm BMO}-$spaces. The latter implies solutions are of class $C^{0,{\rm Log-Lip}}$ across the interface. Relaxing the assumptions on the data, we establish local Hölder continuity for the solutions.

math.AP↗
arXiv2023

Transmission problems: regularity theory, interfaces and beyond

Modelling diffusion processes in heterogeneous media requires addressing inherent discontinuities across interfaces, where specific conditions are to be met. These challenges fall under the purview of Mathematical Analysis as \emph{transmission problems}. We present a pa\-no\-ra\-ma of the theory of transmission problems, encompassing the seminal contributions from the 1950s and subsequent developments. Then we delve into the discussion of regularity issues, including recent advances matching the minimal regularity requirements of interfaces and the optimal regularity of the solutions. A discussion on free transmission problems closes the survey.

math.AP↗
arXiv2023

Improved regularity for a Hessian-dependent functional

We prove that minimizers of the $L^{d}$-norm of the Hessian in the unit ball of $\mathbb{R}^d$ are locally of class $C^{1,α}$. Our findings extend previous results on Hessian-dependent functionals to the borderline case and resonate with the Hölder regularity theory available for elliptic equations in double-divergence form.

math.AP↗
arXiv2022

Flipping regularity via the Harnack approach and applications to nonlinear elliptic problems

We prove an abstract result ensuring that one-sided geometric control yields two-sided estimates for functions satisfying general conditions. Our findings resonate in the context of nonlinear elliptic problems, including supersolutions to fully nonlinear elliptic equations and functions in the De Giorgi class. Among the consequences of our abstract results are regularity estimates, and conditions for a continuous function to be in the class of viscosity solutions. We also prove that one-sided geometric control yields $L^pL^\infty$-estimates. It provides a converse to the implication in the De Giorgi-Nash-Moser theory.

math.AP↗
arXiv2022

Fully nonlinear Hamilton-Jacobi equations of degenerate type

We examine Hamilton-Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii-Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

math.AP↗
arXiv2022

Potential estimates for fully nonlinear elliptic equations with bounded ingredients

We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We briefly survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from -- and are inspired by -- fundamental facts in the theory of $L^p$-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi and Giuseppe Mingione [10].

math.AP↗
arXiv2022

$C^1$-regularity for degenerate diffusion equations

We prove that any solution of a degenerate elliptic PDE is of class $C^1$, provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. $σ^{-1} \in L^1\left (\frac{1}λ {\bf d}λ\right )$. The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate $u(x)$, near $x_0$, by an error of order $\text{o}(|x-x_0|)$. Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the ${C}^1$--regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for the renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate.

math.AP↗
arXiv2022

Fractional Sobolev regularity for fully nonlinear elliptic equations

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only on ellipticity constants and dimension, such that if $u$ is a viscosity solution of $F(D^2u) = f(x) \in L^p$, then $u\in W^{1+\varepsilon,p}$, with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that $F(D^2u) \in L^p \implies (-Δ)^θu \in L^p$, for a universal constant $\frac{1}{2} < θ<1$. We believe our techniques are flexible and can be adapted to various models and contexts.

math.AP↗
arXiv2021

A fully nonlinear degenerate free transmission problem

We study a free transmission problem driven by degenerate fully nonlinear operators. Our first result concerns the existence of solutions to the associated Dirichlet problem. By framing the equation in the context of viscosity inequalities, we produce optimal regularity results for viscosity solutions and certain strong solutions to the problem. Our findings include regularity in $ C^{1,α}$ spaces, and an explicit characterization of $α$ in terms of the degeneracy rates. We argue by perturbation methods, relating our problem to a homogeneous, fully nonlinear uniformly elliptic equation.

math.AP↗
arXiv2021

Existence of solutions to a fully nonlinear free transmission problem

We study an equation governed by a discontinuous fully nonlinear operator. Such discontinuities are solution-dependent, which introduces a free boundary. Working under natural assumptions, we prove the existence of $L^p$-viscosity and strong solutions to the problem. The operator does not satisfy the usual structure conditions and to obtain the existence of solutions we resort to solving approximate problems, combined with an iterative scheme. We believe our strategy can be applied to other classes of non-variational free boundary problems.

math.AP↗
arXiv2020

A fully nonlinear free transmission problem

We examine a free transmission problem driven by fully nonlinear elliptic operators. Since the transmission interface is determined endogeneously, our analysis is two-fold: we study the regularity of the solutions and some geometric properties of the free boundary. By relating our problem with a pair of viscosity inequalities, we prove that strong solutions are of class $C ^{1,1}$, locally. As regards the free boundary, we start by establishing weak results, such as its non-degeneracy, and proceed with the characterization of global solutions.

math.AP↗
arXiv2020

Stationary fully nonlinear mean-field games

In this paper we examine fully nonlinear mean-field games associated with a minimization problem. The variational setting is driven by a functional depending on its argument through its Hessian matrix. We work under fairly natural conditions and establish improved (sharp) regularity for the solutions in Sobolev spaces. Then, we prove the existence of minimizers for the variational problem and the existence of solutions to the mean-field games system. We also investigate a unidimensional example and unveil new information on the explicit solutions. Our findings can be generalized to a larger class of operators, yielding information on a broader range of examples.

math.AP↗