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Félix del Teso

Publications and source records attributed to Félix del Teso.

At least 19 recordsLinked to original sources
arXiv2026

A Viscosity Framework for Dynamic Programming Principles and Applications

In this work we introduce a viscosity-based notion of solution for general approximation schemes associated with partial differential equations, such as dynamic programming principles~(DPPs). A key feature of our approach is that it bypasses any measurability requirement on solutions of the DPP, an assumption that is often difficult to verify and may even fail in relevant examples. We establish a comparison principle between classical strict supersolutions and viscosity subsolutions of the DPP, which yields stability results under minimal and natural hypotheses. As a consequence, we prove existence of viscosity solutions of the DPP and their convergence to viscosity solutions of a PDE that is consistent with the underlying approximation scheme. Moreover, we show that solutions of the limiting PDE admit an asymptotic expansion encoded by the approximation operator. Finally, we demonstrate that a broad class of local, nonlocal, and nonlinear partial differential equations fits into our framework, recovering known examples in the literature and completing gaps in the existing literature.

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arXiv2025

A convergent finite difference-quadrature scheme for the porous medium equation with nonlocal pressure

We introduce and analyze a numerical approximation of the porous medium equation with fractional potential pressure introduced by Caffarelli and Vázquez: \[ \partial_t u = \nabla \cdot (u^{m-1}\nabla (-Δ)^{-σ}u) \qquad \text{for} \qquad m\geq2 \quad \text{and} \quad σ\in(0,1). \] Our scheme is for one space dimension and positive solutions $u$. It consists of solving numerically the equation satisfied by $v(x,t)=\int_{-\infty}^xu(x,t)dx$, the quasilinear non-divergence form equation \[ \partial_t v= -|\partial_x v|^{m-1} (- Δ)^{s} v \qquad \text{where} \qquad s=1-σ, \] and then computing $u=v_x$ by numerical differentiation. Using upwinding ideas in a novel way, we construct a new and simple, monotone and $L^\infty$-stable, approximation for the $v$-equation, and show local uniform convergence to the unique discontinuous viscosity solution. Using ideas from probability theory, we then prove that the approximation of $u$ converges weakly-$*$, or more precisely, up to normalization, in $C(0,T; P(\mathbb{R}))$ where $P(\mathbb{R})$ is the space of probability measures under the Rubinstein-Kantorovich metric.The analysis include also fundamental solutions where the initial data for $u$ is a Dirac mass. Numerical tests are included to confirm the results. Our scheme seems to be the first numerical scheme for this type of problems.

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arXiv2025

Fractional Derivatives: an extension of classical analysis to non-integer orders

This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant properties and applications of this mathematical tool. It was originally published in Spanish in the Gaceta de la Real Sociedad Española and automatically translated using Github copilot.

math.CA↗
arXiv2025

Finite Elements with weighted bases for the fractional Laplacian

This work presents a numerical study of the Dirichlet problem for the fractional Laplacian $(-Δ)^s$ with $s\in(0,1)$ using Finite Element methods with non-standard bases. Classical approaches based on piece-wise linear basis yield $h^{\frac 1 2}$ convergence rates in the Sobolev-Slobodeckij norm $H^s$ due to the limited boundary regularity of the solution $u(x)$, which behaves like $\operatorname{dist}(x,\mathbb{R}^d\setminus Ω)^s$, where $h$ is the diameter of the mesh elements. To overcome this limitation, we propose a novel Finite Element basis of the form $δ^s \times ($piece-wise linear functions$)$, where $δ$ is any suitably smooth approximation of $\operatorname{dist}(x,\mathbb{R}^d\setminus Ω)$. This exploits the improved regularity of $u/δ^s$, achieving higher convergence rates. Under standard smoothness assumptions the method attains an order $h^{2-s}$ on quasi-uniform meshes, improving the rates with the piece-wise linear basis. We provide a rigorous theoretical error analysis with explicit rates and validate it through numerical experiments.

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arXiv2025

The Fujita exponent for finite difference approximations of nonlocal and local semilinear blow-up problems

We study monotone finite difference approximations for a broad class of reaction-diffusion problems, incorporating general symmetric Lévy operators. By employing an adaptive time-stepping discretization, we derive the discrete Fujita critical exponent for these problems. Additionally, under general consistency assumptions, we establish the convergence of discrete blow-up times to their continuous counterparts. As complementary results, we also present the asymptotic-in-time behavior of discrete heat-type equations as well as an extensive analysis of discrete eigenvalue problems.

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arXiv2024

Game theoretical asymptotic mean value properties for non-homogeneous $p$-Laplace problems

We extend the classical mean value property for the Laplacian operator to address a nonlinear and non-homogeneous problem related to the $p$-Laplacian operator for $p>2$. Specifically, we characterize viscosity solutions to the $p$-Laplace equation $Δ_p u:=\nabla\cdot(|\nabla u|^{p-2} \nabla u) = f$ with a nontrivial right-hand side $f$, through novel asymptotic mean value formulas. While asymptotic mean value formulas for the homogeneous case ($f = 0$) have been previously established, leveraging the normalization $Δ_p^{\text{N}}u:=|\nabla u|^{2-p} Δ_p u = 0$, which yields the 1-homogeneous normalized $p$-Laplacian, such normalization is not applicable when $f \neq 0$. Furthermore, the mean value formulas introduced here motivate, for the first time in the literature, a game-theoretical approach for non-homogeneous $p$-Laplace equations. We also analyze the existence, uniqueness, and convergence of the game values, which are solutions to a dynamic programming principle derived from the mean value property.

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arXiv2024

Convergence rates of the fractional to the local Dirichlet problem

We prove non-asymptotic rates of convergence in the $W^{s,2}(\mathbb R^d)$-norm for the solution of the fractional Dirichlet problem to the solution of the local Dirichlet problem as $s\uparrow 1$. For regular enough boundary values we get a rate of order $\sqrt{1-s}$, while for less regular data the rate is of order $\sqrt{(1-s)|\log(1-s)|}$. We also obtain results when the right hand side depends on $s$, and our error estimates are true for all $s\in(0,1)$. The proofs use variational arguments to deduce rates in the fractional Sobolev norm from energy estimates between the fractional and the standard Dirichlet energy.

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arXiv2023

Numerical methods and regularity properties for viscosity solutions of nonlocal in space and time diffusion equations

We consider a general family of nonlocal in space and time diffusion equations with space-time dependent diffusivity and prove convergence of finite difference schemes in the context of viscosity solutions under very mild conditions. The proofs, based on regularity properties and compactness arguments on the numerical solution, allow to inherit a number of interesting results for the limit equation. More precisely, assuming Hölder regularity only on the initial condition, we prove convergence of the scheme, space-time Hölder regularity of the solution depending on the fractional orders of the operators, as well as specific blow up rates of the first time derivative. Finally, using the obtained regularity results, we are able to prove orders of convergence of the scheme in some cases. These results are consistent with previous studies. The schemes' performance is further numerically verified using both constructed exact solutions and realistic examples. Our experiments show that multithreaded implementation yields an efficient method to solve nonlocal equations numerically.

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arXiv2023

Higher-order asymptotic expansions and finite difference schemes for the fractional $p$-Laplacian

We propose a new asymptotic expansion for the fractional $p$-Laplacian with precise computations of the errors. Our approximation is shown to hold in the whole range $p\in(1,\infty)$ and $s\in(0,1)$, with errors that do not degenerate as $s\to1^-$. These are super-quadratic for a wide range of $p$ (better far from the zero gradient points), and optimal in most cases. One of the main ideas here is the fact that the singular part of the integral representation of the fractional $p$-Laplacian behaves like a local $p$-Laplacian with a weight correction. As a consequence of this, we also revisit a previous asymptotic expansion for the classical $p$-Laplacian, whose error orders were not known. Based on the previous result, we propose monotone finite difference approximations of the fractional $p$-Laplacian with explicit weights and we obtain the error estimates. Finally, we introduce explicit finite difference schemes for the associated parabolic problem in $\mathbb{R}^d$ and show that it is stable, monotone and convergent in the context of viscosity solutions. An interesting feature is the fact that the stability condition improves with the regularity of the initial data.

math.AP↗
arXiv2022

Evolution Driven by the Infinity Fractional Laplacian

We consider the evolution problem associated to the infinity fractional Laplacian introduced by Bjorland, Caffarelli and Figalli (2012) as the infinitesimal generator of a non-Brownian tug-of-war game. We first construct a class of viscosity solutions of the initial-value problem for bounded and uniformly continuous data. An important result is the equivalence of the nonlinear operator in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially symmetric and monotone functions. Thanks to this and a comparison theorem between classical and viscosity solutions, we are able to establish a global Harnack inequality that, in particular, explains the long-time behavior of the solutions.

math.AP↗
arXiv2022

The one-phase fractional Stefan problem

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in $\mathbb{R}^N$. In terms of the enthalpy $h(x,t)$, the evolution equation reads $\partial_t h+(-Δ)^sΦ(h) =0$, while the temperature is defined as $u:=Φ(h):=\max\{h-L,0\}$ for some constant $L>0$ called the latent heat, and $(-Δ)^s$ stands for the fractional Laplacian with exponent $s\in(0,1)$. We prove the existence of a continuous and bounded selfsimilar solution of the form $h(x,t)=H(x\,t^{-1/(2s)})$ which exhibits a free boundary at the change-of-phase level $h(x,t)=L$. This level is located at the line (called the free boundary) $x(t)=ξ_0 t^{1/(2s)}$ for some $ξ_0>0$. The construction is done in 1D, and its extension to $N$-dimensional space is shown. We also provide well-posedness and basic properties of very weak solutions for general bounded data $h_0$ in several dimensions. The temperatures $u$ of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of $u$ for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for $u$ so that the support never recedes. On the contrary, the enthalpy $h$ has infinite speed of propagation and we obtain precise estimates on the tail. The limits $L\to0^+$, $L\to +\infty$, $s\to0^+$ and $s\to 1^-$ are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena.

math.AP↗
arXiv2022

On asymptotic expansions for the fractional infinity Laplacian

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional $\infty$-Laplacian $Δ_\infty^s$ for $s\in (\frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland, C., Caffarelli, L. and Figalli, A., \textsl{Nonlocal Tug-of-War and the inifnity fractional Laplacian}, Comm. Pure Appl. Math., \textbf{65}, pp. 337--380, (2012)]. Our expansions are parametrised by the radius of the removed singularity $ε$, and allow for the identification of $Δ_\infty^sϕ(x)$ as the $ε^{2s}$-order coefficient of the deviation of the $ε$-average from the value $ϕ(x)$, in the limit $ε\to 0+$. The averages are well posed for functions $ϕ$ that are only Borel regular and bounded.

math.AP↗
arXiv2022

Uniform tail estimates and $L^p(\mathbb{R}^N)$-convergence for finite-difference approximations of nonlinear diffusion equations

We obtain new equitightness and $C([0,T];L^p(\mathbb{R}^N))$-convergence results for finite-difference approximations of generalized porous medium equations of the form $$ \partial_tu-\mathfrak{L}[φ(u)]=g\qquad\text{in $\mathbb{R}^N\times(0,T)$}, $$ where $φ:\mathbb{R}\to\mathbb{R}$ is continuous and nondecreasing, and $\mathfrak{L}$ is a local or nonlocal diffusion operator. Our results include slow diffusions, strongly degenerate Stefan problems, and fast diffusions above a critical exponent. These results improve the previous $C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$-convergence obtained in a series of papers on the topic by the authors. To have equitightness and global $L^p(\mathbb{R}^N)$-convergence, some additional restrictions on $\mathfrak{L}$ and $φ$ are needed. Most commonly used symmetric operators $\mathfrak{L}$ are still included: the Laplacian, fractional Laplacians, and other generators of symmetric Lévy processes with some fractional moment. We also discuss extensions to nonlinear possibly strongly degenerate convection-diffusion equations.

math.AP↗
arXiv2022

Finite difference schemes for the parabolic $p$-Laplace equation

We propose a new finite difference scheme for the degenerate parabolic equation \[ \partial_t u - \mbox{div}(|\nabla u|^{p-2}\nabla u) =f, \quad p\geq 2. \] Under the assumption that the data is Hölder continuous, we establish the convergence of the explicit-in-time scheme for the Cauchy problem provided a suitable stability type CFL-condition. An important advantage of our approach, is that the CFL-condition makes use of the regularity provided by the scheme to reduce the computational cost. In particular, for Lipschitz data, the CFL-condition is of the same order as for the heat equation and independent of $p$.

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arXiv2021

Three representations of the fractional $p$-Laplacian: semigroup, extension and Balakrishnan formulas

We introduce three representation formulas for the fractional $p$-Laplace operator in the whole range of parameters $0<s<1$ and $1<p<\infty$. Note that for $p\ne 2$ this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre linear extension technique. The third one is the corresponding nonlinear version of the Balakrishnan formula. We also discuss the correct choice of the constant of the fractional $p$-Laplace operator in order to have continuous dependence as $p\to 2$ and $s \to 0^+, 1^-$. A number of consequences and proposals are derived. Thus, we propose a natural spectral-type operator in domains, different from the standard restriction of the fractional $p$-Laplace operator acting on the whole space. We also propose numerical schemes, a new definition of the fractional $p$-Laplacian on manifolds, as well as alternative characterizations of the $W^{s,p}(\mathbb{R}^n)$ seminorms.

math.AP↗
arXiv2021

A mean value formula for the variational $p$-Laplacian

We prove a new asymptotic mean value formula for the $p$-Laplace operator, $$ Δ_p u=\text{div}(|\nabla u|^{p-2}\nabla u), $$ valid in the viscosity sense. In the plane, and for a certain range of $p$, the mean value formula holds in the pointwise sense. We also study the existence, uniqueness and convergence of the related dynamic programming principle.

math.AP↗
arXiv2021

A finite difference method for the variational $p$-Laplacian

We propose a new monotone finite difference discretization for the variational $p$-Laplace operator, \[ Δ_p u=\text{div}(|\nabla u|^{p-2}\nabla u), \] and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational $p$-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

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