arXiv subjects

Davide Azevedo

Publications and source records attributed to Davide Azevedo.

7 recordsLinked to original sources

arXiv2025

Limits of sequences of volume preserving homeomorphisms in $W^{1,p}$, for $0<p<1$

If $Ω$ is an open subset of $\mathbb{R}$ and $p>0$ then the elements of $W^{1,p}(Ω)$ can be seen as the pairs $(f,F)\in L^p(Ω)\times (L^p(Ω))^d$ such that there exists a sequence $(f_n)_n$ of $C^1$ functions converging to $f$ in $L^p(Ω)$ such that $(\nabla f_n)_n$ converges to $F$ in $(L^p(Ω))^d$. If $p\geq 1$ the pair $(f,F)$ is defined by $f$ as $F$ must be the distributional gradient of $f$. If $0 1$, admits a sequence $(f_n)_n$ of $C^1$ homeomorphisms uniformly converging to $f$ and such that $(f_n')_n$ converges in $L^p(I)$ to $F$, if and only if $0\leq \frac{F}{f'}\leq 1$.

math.DS↗

arXiv2025

Lagrange multipliers and characteristic functions

We consider a stationary variational inequality with gradient constraint and obstacle. We prove that this problem can be described by an equation using a Lagrange multiplier and a characteristic function. The Lagrange multiplier contains information about the contact set of the modulus of the gradient of the solution with the gradient constraint, and the characteristic function is defined in the contact set of the solution with the obstacle. Moreover, given a convergent sequence of data, we prove the stability of the corresponding solutions.

math.AP↗

arXiv2025

An evolutionary vector-valued variational inequality and Lagrange multiplier

We prove existence and uniqueness of solution of an evolutionary vector-valued variational inequality defined in the convex set of vector valued functions $\boldsymbol v$ subject to the constraint $|\boldsymbol v|\le1$. We show that we can write the variational inequality as a system of equations on the unknowns $(λ,\boldsymbol u)$, where $λ$ is a (unique) Lagrange multiplier belonging to $L^p$ and $\boldsymbol u$ solves the variational inequality. Given data $(\boldsymbol f_n,\boldsymbol u_{n0})$ converging to $(\boldsymbol f,\boldsymbol u_0)$ in $\boldsymbol L^\infty(Q_T)\times H^1_0(Ω)$, we prove the convergence of the solutions $(λ_n,\boldsymbol u_n)$ of the Lagrange multiplier problem to the solution of the limit problem, when we let $n\rightarrow \infty$.

math.AP↗

arXiv2025

Genericity of ergodicity for Sobolev homeomorphisms

In this paper we obtain a weak version of Lusin's theorem in the Sobolev-$(1,p)$ uniform closure of volume preserving Lipschitz homeomorphisms on closed and connected $d$-dimensional manifolds, $d \geq 2$ and $0<p<1$. With this result at hand we will be able to prove that the ergodic elements are generic. This establishes a version of Oxtoby and Ulam theorem for this Sobolev class. We also prove that, for $1\leq p<d-1$, the topological transitive maps are generic.

math.DS↗

arXiv2016

Extreme Value Laws for dynamical systems with countable extremal sets

We consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system. We focus on the extremal behaviour of the process, which is related to the entrance in certain regions of the phase space, which correspond to neighbourhoods of the maximal set $\mathcal M$, i.e. the set of points where the observable is maximised. The main novelty here is the fact that we consider that the set $\mathcal M$ may have a countable number of points, which are associated by belonging to the orbit of a certain point, and may have accumulation points. In order to prove the existence of distributional limits and study the intensity of clustering, given by the Extremal Index, we generalise the conditions previously introduced in \cite{FFT12,FFT15}.

math.DS↗

arXiv2015

Clustering of extreme events created by multiple correlated maxima

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The novelty is that we will consider observables achieving a global maximum value (possible infinite) at multiple points with special emphasis for the case where these maximal points are correlated or bound by belonging to the same orbit of a certain chosen point. These multiple correlated maxima can be seen as a new mechanism creating clustering. We recall that clustering was intimately connected with periodicity when the maximum was achieved at a single point. We will study this mechanism for creating clustering and will address the existence of limiting Extreme Value Laws, the repercussions on the value of the Extremal Index, the impact on the limit of Rare Events Points Processes, the influence on clustering patterns and the competition of domains of attraction. We also consider briefly and for comparison purposes multiple uncorrelated maxima. The systems considered include expanding maps of the interval such as Rychlik maps but also maps with an indifferent fixed point such as Manneville-Pommeau maps.

math.DS↗

arXiv2013

Statistical properties of diffeomorfisms with weak invariant manifolds

We consider diffeomorphisms of compact Riemmanian manifolds which have a Gibbs-Markov-Young structures, consisting of a reference set $Λ$ with a hyperbolic product structure and a countable Markov partition. We assume polynomial contraction on stable leaves, polynomial backward contraction on unstable leaves, a bounded distortion property and a certain regularity of the stable foliation. We establish a control on the decay of correlations and large deviations of the SRB measure of the dynamical system, based on a polynomial control on the Lebesgue measure of the tail of return times. Finally, we present an example of a dynamical system defined on the torus and prove that it verifies the properties of the Gibbs-Markov-Young structure that we considered.

math.DS↗