arXiv subjects

Fernando Charro

Publications and source records attributed to Fernando Charro.

15 recordsLinked to original sources

arXiv2026

A geometric approach to nonlocal 2-Hessian equations

We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}\Delta^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range $s\in(0,1)$. In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as $s\to1$, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.

math.AP↗

arXiv2026

On the characterization of polyharmonic functions through iterated means

We introduce an infinite family of mean-value formulas (exact and asymptotic) given in terms of linear combinations of iterated means. We prove that the mean-value formulas in this family characterize real-valued polyharmonic functions of finite order, and that a simple algebraic condition partitions the family into equivalence classes according to the order of polyharmonicity. Our key results include strong converses to the mean-value properties -- locally integrable functions satisfying a mean-value property in the family are polyharmonic -- and a regularity result -- locally integrable functions satisfying a mean-value property in the family, whether exact or asymptotic, are smooth.

math.AP↗

arXiv2026

Pointwise mean-value formulas with quantitative remainder for higher-order Poisson equations

Higher-order Poisson equations involve an integer power of the Laplacian and a nonzero forcing term, and appear in areas of physics and engineering such as hydrodynamics, structural engineering, and image processing. We introduce a family of mean-value formulas for solutions to higher-order Poisson equations, given in terms of linear combinations of iterated means, with an exact remainder quantified by the oscillation of the forcing term. We also prove a regularity result and a strong converse to the mean-value property, in the sense that merely locally integrable functions satisfying our formulas are regular solutions of the higher-order Poisson equation. In contrast with the homogeneous case, where the mean-value property forces smoothness, the regularity attainable here is dictated by the regularity of the forcing term. Together, our results provide a mean-value characterization of solutions to higher-order Poisson equations.

math.AP↗

arXiv2021

The Gelfand problem for the Infinity Laplacian

We study the asymptotic behavior as $p\to\infty$ of the Gelfand problem \[ -\Delta_{p} u=\lambda\,e^{u}\ \textrm{in}\ \Omega\subset\mathbb{R}^n,\quad u=0 \ \textrm{on}\ \partial\Omega. \] Under an appropriate rescaling on $u$ and $\lambda$, we prove uniform convergence of solutions of the Gelfand problem to solutions of \[ \min\left\{|\nabla{}u|-\Lambda\,e^{u}, -\Delta_{\infty}u\right\}=0\ \textrm{in}\ \Omega,\quad u=0\ \text{on}\ \partial\Omega. \] We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of $\Lambda$.

math.AP↗

arXiv2021

Asymptotic Mean-Value Formulas for Solutions of General Second-Order Elliptic Equations

We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and $k$-Hessian operators.

math.AP↗

arXiv2021

Asymptotic mean value formulas for parabolic nonlinear equations

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Amp\`ere equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of Dynamic Programming Principles for certain two-player, zero-sum games.

math.AP↗

arXiv2020

A Nonlinear Mean Value Property for Monge-Amp\`ere

In recent years there has been an increasing interest in whether a mean value property, known to characterize harmonic functions, can be extended in some weak form to solutions of nonlinear equations. This question has been partially motivated by the surprising connection between Random Tug-of-War games and the normalized $p-$Laplacian discovered some years ago, where a nonlinear asymptotic mean value property for solutions of a PDE is related to a dynamic programming principle for an appropriate game. Currently, asymptotic nonlinear mean value formulas are rare in the literature and our goal is to show that an asymptotic nonlinear mean value formula holds for the classical Monge-Amp\`ere equation.

math.AP↗

arXiv2019

The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space

We study the following natural question that, apparently, has not been well addressed in the literature: Given functions $u$ with support in the unit ball $B_1\subset\mathbb{R}^n$ and with gradient in the Morrey space $M^{p,\lambda}(B_1)$, where $1 \lambda p/(\lambda-p)$. The function is basically a negative power of the distance to a set of Hausdorff dimension $n-\lambda$. When $\lambda\notin\mathbb{Z}$, this set is a fractal. We also make a detailed study of the radially symmetric case, a situation in which the exponent $q$ can go up to $np/(\lambda-p)$.

math.AP↗

arXiv2019

Explicit solutions of Jensen's auxiliary equations via extremal Lipschitz extensions

In this note we prove that McShane and Whitney's Lipschitz extensions are viscosity solutions of Jensen's auxiliary equations, known to have a key role in Jensen's celebrated proof of uniqueness of infinity harmonic functions, and therefore of Absolutely Minimizing Lipschitz Extensions. To the best of the author's knowledge, this result does not appear to be known in the literature in spite of the vast amount of work around the topic.

math.AP↗

arXiv2019

Totalitarian random Tug-of-War games in graphs

In this work we discuss a random Tug-of-War game in graphs where one of the players has the power to decide at each turn whether to play a round of classical random Tug-of-War, or let the other player choose the new game position in exchange of a fixed payoff. We prove that this game has a value using a discrete comparison principle and viscosity tools, as well as probabilistic arguments. This game is related to Jensen's extremal equations, which have a key role in Jensen's celebrated proof of uniqueness of infinity harmonic functions.

math.AP↗

arXiv2014

On the existence threshold for positive solutions of p-laplacian equations with a concave-convex nonlinearity

We study the following boundary value problem with a concave-convex nonlinearity: \begin{equation*} \left\{ \begin{array}{r c l l} -Δ_p u & = & Λ\,u^{q-1}+ u^{r-1} & \textrm{in }Ω, \\ u & = & 0 & \textrm{on }\partialΩ. \end{array}\right. \end{equation*} Here $Ω\subset \mathbb{R}^n$ is a bounded domain and $1 0$ such that the problem admits at least two positive solutions for $0<Λ<Λ_{q,r}$, at least one positive solution for $Λ=Λ_{q,r}$, and no positive solution for $Λ> Λ_{q,r}$. We show that \[ \lim_{q \to p} Λ_{q,r} = λ_1(p), \] where $λ_1(p)$ is the first eigenvalue of the p-laplacian. It is worth noticing that $λ_1(p)$ is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case $q=p$.

math.AP↗

arXiv2014

An existence result for the infinity laplacian with non-homogeneous Neumann boundary conditions using Tug-of-War games

In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with non-homogeneous mixed boundary conditions. For a Lipschitz and positive function $g$ there exists a viscosity solution of the mixed boundary value problem, $$ \{\begin{array}{ll} \displaystyle -Δ_{\infty}u(x)=0\quad & \text{in} Ω, \displaystyle \frac{\partial u}{\partial n}(x)= g (x)\quad & \text{on} Γ_N, \displaystyle u(x)= 0 \quad & \text{on} Γ_D. \end{array}. $$

math.AP↗

arXiv2014

On a fractional Monge-Amp\`ere operator

In this paper we consider a fractional analogue of the Monge-Amp\`ere operator. Our operator is a concave envelope of fractional linear operators of the form $ \inf_{A\in \mathcal{A}}L_Au, $ where the set of operators corresponds to all affine transformations of determinant one of a given multiple of the fractional Laplacian. We set up a relatively simple framework of global solutions prescribing data at infinity and global barriers. In our key estimate, we show that the operator remains strictly elliptic, which allows to apply known regularity results for uniformly elliptic operators and deduce that solutions are classical.

math.AP↗

arXiv2011

On the Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian

We prove $L^\infty$ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and $p$-Laplacian, namely \[ -Δ_p^N u=f\qquad\text{for $n<p\leq\infty$.} \] We are able to provide a stable family of results depending continuously on the parameter $p$. We also prove the failure of the classical Alexandrov-Bakelman-Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.

math.AP↗

arXiv2009

A mixed problem for the infinity laplacian via Tug-of-War games

In this paper we prove that a function $ u\in\mathcal{C}(\barΩ)$ is the continuous value of the Tug-of-War game described in \cite{PSSW} if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions {-Δ_{\infty}u(x)=0\quad & \text{in} Ω, \frac{\partial u}{\partial n}(x)=0\quad & \text{on} Γ_N, u(x)=F(x)\quad & \text{on} Γ_D. By using the results in \cite{PSSW}, it follows that this viscous PDE problem has a unique solution, which is the unique {\it absolutely minimizing Lipschitz extension} to the whole $\barΩ$ (in the sense of \cite{Aronsson} and \cite{PSSW}) of the boundary data $ F:Γ_D\to\R $.

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