arXiv subjects

Felipe Ponce-Vanegas

Publications and source records attributed to Felipe Ponce-Vanegas.

6 recordsLinked to original sources

arXiv2022

Pointwise convergence over fractals for dispersive equations with homogeneous symbol

We study the fractal pointwise convergence for the equation $i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ with $s>(n-α+1)/2$ the solution $u$ converges to $f$ $\mathcal{H}^α$-a.e, where $\mathcal{H}^α$ is the $α$-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of $P$. On the other hand, for a family of polynomials $P$ and given $α$, we exploit a Talbot-like effect to construct initial data whose solutions $u$ diverge in sets of Hausdorff dimension $α$. To compute the dimension of the sets of divergence, we adopt the Mass Transference Principle from Diophantine approximation. We also construct counterexamples for quadratic symbols like the saddle to show that our positive results are sometimes best possible.

math.AP↗

arXiv2022

Static and Dynamical, Fractional Uncertainty Principles

We study the process of dispersion of low-regularity solutions to the Schrödinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional weights and use it to get a lower bound for the concentration of mass. We consider also the evolution when the initial datum is the Dirac comb in $\mathbb{R}$. In this case we find fluctuations that concentrate at rational times and that resemble a realization of a Lévy process. Furthermore, the evolution exhibits multifractality.

math.AP↗

arXiv2021

Counterexamples for the fractal Schrödinger convergence problem with an intermediate space trick

We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same regularity as Du's counterexamples for weighted $L^2$ restriction estimates is achieved for the convergence problem. To do so, we need to construct the set of divergence explicitly and compute its Hausdorff dimension, for which we use the Mass Transference Principle, a technique originated from Diophantine approximation.

math.AP↗

arXiv2021

Convergence over fractals for the Schrödinger equation

We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the $α$-Hausdorff measure ($α$-a.e.). We extend to the fractal setting ($α< n$) a recent counterexample of Bourgain \cite{Bourgain2016}, which is sharp in the Lebesque measure setting ($α= n$). In doing so we recover the necessary condition from \cite{zbMATH07036806} for pointwise convergence~$α$-a.e. and we extend it to the range $n/2<α\leq (3n+1)/4$.

math.AP↗

arXiv2019

Recovery of the Derivative of the Conductivity at the Boundary

We describe a method to reconstruct the conductivity and its normal derivative at the boundary from the knowledge of the potential and current measured at the boundary. This boundary determination implies the uniqueness of the conductivity in the bulk when it lies in $W^{1+\frac{n-5}{2p}+,p}$, for dimensions $n\ge 5$ and for $n\le p<\infty$.

math.AP↗

arXiv2019

The Bilinear Strategy for Calderón's Problem

Electrical Impedance Imaging would suffer a serious obstruction if for two different conductivities the potential and current measured at the boundary were the same. The Calderón's problem is to decide whether the conductivity is indeed uniquely determined by the data at the boundary. In $\mathbb{R}^d$, for $d=5,6$, we show that uniqueness holds when the conductivity is in $W^{1+\frac{d-5}{2p}+, p}(Ω)$, for $d\le p <\infty$. This improves on recent results of Haberman, and of Ham, Kwon and Lee. The main novelty of the proof is an extension of Tao's bilinear Theorem.

math.AP↗