arXiv subjects

Sebastián Tapia-García

Publications and source records attributed to Sebastián Tapia-García.

12 recordsLinked to original sources

arXiv2026

Sharp regularity results for eikonal equations in Banach spaces

Viscosity solutions of eikonal equations need not enjoy regularity beyond Lipschitz continuity. In contrast, differentiable solutions exhibit stronger structural properties. In the Euclidean setting, building on observations of Caffarelli and Crandall (Comm. Partial Differential Equations $\textbf{35}$ (2010), 391--414), one finds that differentiable solutions of the eikonal equation $\| Df \| \equiv 1 \,\, \text{in } \mathcal{U} \subset \mathbb{R}^d$ are of class $C^{1, 1}_{\mathrm{loc}}(\mathcal U)$. Moreover, if $\mathcal{U} = \mathbb{R}^d$, then $f$ must be affine. This manuscript revisits these phenomena in the setting of (finite-- and infinite--dimensional) Banach spaces. Our results show how the geometry of the underlying Banach space determines the regularity and affineness of Gateaux--differentiable solutions of eikonal equations. As a geometric application in finite dimensions, we characterize differentiable functions whose gradient ranges are constrained to boundaries of convex bodies.

math.AP↗

arXiv2026

Invariant subspaces for free linearizations of Lipschitz maps

We consider the invariant subspace problem for operators induced by Lipschitz self-maps on Lipschitz-free spaces. Besides the usual free linearizations $\widehat f$ of basepoint-preserving Lipschitz self-maps, we consider a wider class of operators $T_{f,e}$ which are naturally defined for arbitrary Lipschitz self-maps. We show, among other results, that every $\widehat f$ admits a non-trivial invariant subspace whenever the underlying metric space contains a compact ball, or has at least two connected components one of which has non-empty interior. We also obtain corresponding positive results for $T_{f,e}$ under isolated-point, compact-ball and disconnectedness assumptions, and discuss some consequences for linear dynamics.

math.FA↗

arXiv2026

The reach and limits of slope eikonal equations in compact spaces

It is a well known fact that the eikonal equation is well posed in complete length spaces. Among the studied notions of solutions in the literature, there is one that can be defined in any metric space using the local (descent) slope and considering pointwise solutions: functionals such that their slope coincides with the prescribed data at every point of the domain. In this work we explore the question ``Can we characterize the class of compact metric spaces in which every slope eikonal equation (under standard assumptions) always admits a pointwise solution?''. We provide a purely metric characterization of these spaces, as well as some interesting examples and counterexamples that illustrate the reach and limitations of the concept.

math.FA↗

arXiv2025

Determination of (unbounded) convex functions via Crandall-Pazy directions

It has been recently discovered that a convex function can be determined by its slopes and its infimum value, provided this latter is finite. The result was extended to nonconvex functions by replacing the infimum value by the set of all critical and asymptotically critical values. In all these results boundedness from below plays a crucial role and is generally admitted to be a paramount assumption. Nonetheless, this work develops a new technique that allows to also determine a large class of unbounded from below convex functions, by means of a Neumann-type condition related to the Crandall-Pazy direction.

math.FA↗

arXiv2024

On (discounted) global Eikonal equations in metric spaces

Eikonal equations in metric spaces have strong connections with the local slope operator (or the De Giorgi slope). In this manuscript, we explore and delve into an analogous model based on the global slope operator, expressed as $λu + G[u] = \ell$, where $λ\geq 0$. In strong contrast with the classical theory, the global slope operator relies neither on the local properties of the functions nor on the structure of the space, and therefore new insights are developed in order to analyze the above equation. Under mild assumptions on the metric space $X$ and the given data $\ell$, we primarily discuss: $(a)$ the existence and uniqueness of (pointwise) solutions; $(b)$ a viscosity perspective and the employment of Perron's method to consider the maximal solution; $(c)$ stability of the maximal solution with respect to both, the data $\ell$ and the discount factor $λ$. Our techniques provide a method to approximate solutions of Eikonal equations in metric spaces and a new integration formula based on the global slope of the given function.

math.AP↗

arXiv2023

A slope generalization of Attouch theorem

A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we show that in finite dimensions and under a mild boundedness assumption, we can replace subdifferentials (sets of vectors) by slopes (scalars, corresponding to the distance of the subdifferentials to zero) and still obtain the same characterization: namely, the epigraphical convergence of functions is equivalent to the epigraphical convergence of their slopes. This surprising result goes in line with recent developments on slope determination (Boulmezaoud, Cieutat, Daniilidis, 2018), (Pérez-Aros, Salas, Vilches, 2021) and slope sensitivity (Daniilidis, Drusvyatskiy, 2023) for convex functions.

math.OC↗

arXiv2023

Recurrence and vectors escaping to infinity for Lipschitz operators

We investigate dynamical properties of linear operators that are obtained as the linearization of Lipschitz self-maps defined on a pointed metric space. These operators are known as Lipschitz operators. More concretely, for a Lipschitz operator $\widehat{f}$, we study the set of recurrent vectors and the set of vectors $μ$ such that the sequence $(\|\widehat{f}^n(μ)\|)_n$ goes to infinity. As a consequence of our results we get that there is no wild Lipschitz operator. Furthermore, several examples are presented illustrating our ideas. We highlight the cases when the underlying metric space is a connected subset of $\mathbb{R}$ or a subset of $\mathbb{Z}^d$. We end this paper studying some topological properties of the set of Lipschitz operators.

math.FA↗

arXiv2022

Cyclicity of composition operators on the Fock space

In this paper we provide a full characterization of cyclic composition operators defined on the d-dimensional Fock space $\mathcal F(\mathbb C^d)$ in terms of their symbol. Also, we study the supercyclicity and convex-cyclicity of this type of operators. We end this work by computing the approximation numbers of compact composition operators defined on $\mathcal F(\mathbb C^d)$.

math.FA↗

arXiv2021

Desingularization of the Sweeping Process Mapping

In [9], the celebrated KŁ-inequality has been extended from definable functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R} $ to definable multivalued maps $S:\mathbb{R}\rightrightarrows\mathbb{R}^{n}$, by establishing that the co-derivative mapping $D^{\ast}S$ admits a desingularization around every critical value. As was the case in the gradient dynamics, this desingularization yields a uniform control of the lengths of all bounded orbits of the corresponding sweeping process $-\dotγ(t)\in N_{S(t)}(γ(t))$. In this paper, working outside the framework of o-minimal geometry, we characterize the existence of a desingularization for the coderivative in terms of the behaviour of the sweeping process orbits and the integrability of the talweg function. These results are close in spirit with the ones in [3], where characterizations for the desingularization of the (sub)gradient of functions had been obtained.

math.OC↗

arXiv2021

An epsilon-hypercyclicity criterion and its application on classical Banach spaces

We provide a criterion for $\varepsilon$-hypercyclicity. Also, we extend the ideas of Badea, Grivaux, Müller and Bayart to construct $\varepsilon$-hypercyclic operators which are not hypercyclic in a wider class of separable Banach spaces, including several classical Banach spaces. For instance, our result can be applied to separable infinite dimensional $L^p$ spaces and $C(K)$ spaces.

math.FA↗

arXiv2021

Non-linear operators and differentiability of Lipschitz functions

In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact, limited and completely continuous linear operators. Moreover, our results yield a characterization of Gelfand-Phillips spaces and recover some known result of Schur spaces and reflexive spaces concerning the differentiability of real-valued Lipschitz functions.

math.FA↗

arXiv2019

Compact and Limited Operators

Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider class of functions but still with Lipschitz flavour. As an application we obtain a Banach-Stone-like theorem. On the other hand, we give an extension of a result of Bourgain and Diestel related to limited operators and cosingularity.

math.FA↗