arXiv subjects

Ignacio Ojea

Publications and source records attributed to Ignacio Ojea.

7 recordsLinked to original sources

arXiv2026

The Fractional Korn Inequality on Uniform Domains and New Korn Inequalities for Truncated Seminorms

We prove the so-called second case of the fractional Korn inequality for uniform domains. We obtain this result as an application of a novel fractional Korn-type inequality formulated in terms of truncated seminorms, which turns out to be valid for the broader class of John domains. We also obtain weighted estimates in which the weights are certain powers of the distance to the boundary that depend on the fractional exponent and the Assouad codimension of the boundary of the domain.

math.AP↗

arXiv2024

Poincaré and Sobolev inequalities with variable exponents and log-Holder continuity only at the boundary

We prove Sobolev-Poincaré and Poincaré inequalities in variable Lebesgue spaces $L^{p(\cdot)}(Ω)$, with $Ω\subset{\mathbb R}^n$ a bounded John domain, with weaker regularity assumptions on the exponent $p(\cdot)$ that have been used previously. In particular, we require $p(\cdot)$ to satisfy a new \emph{boundary $\log$-Hölder condition} that imposes some logarithmic decay on the oscillation of $p(\cdot)$ towards the boundary of the domain. Some control over the interior oscillation of $p(\cdot)$ is also needed, but it is given by a very general condition that allows $p(\cdot)$ to be discontinuous at every point of $Ω$. Our results follows from a local-to-global argument based on the continuity of certain Hardy type operators. We provide examples that show that our boundary $\log$-Hölder condition is essentially necessary for our main results. The same examples are adapted to show that this condition is not sufficient for other related inequalities. Finally, we give an application to a Neumann problem for a degenerate $p(\cdot)$-Laplacian.

math.AP↗

arXiv2023

Some inequalities on weighted Sobolev spaces, distance weights and the Assouad dimension

We study certain inequalities and a related result on weighted Sobolev spaces on bounded John domains in $\mathbb{R}^n$. Namely, we prove the existence of a right inverse for the divergence operator, along with the corresponding a priori estimate, the improved and the fractional Poincaré inequalities, the Korn inequality and the local Fefferman-Stein inequality. All these results are obtained on weighted Sobolev spaces, where the weight is a power of the distance to the boundary. In all cases the exponent of the weight $d(\cdot,\partialΩ)^{βp}$ is only required to satisfy the restriction: $βp>-(n-\dim_A(\partialΩ))$, where $p$ is the exponent of the Sobolev space and $\dim_A(\partialΩ)$ is the Assouad dimension of the boundary of the domain. According to our best knowledge, this condition is less restrictive than the ones in the literature.

math.AP↗

arXiv2023

Anisotropic regularity for elliptic problems with Dirac measures as data

We study the Possion problem with singular data given by a source supported on a one dimensional curve strictly contained in a three dimensional domain. We prove regularity results for the solution on isotropic and on anisotropic weighted spaces of Kondratiev type. Our technique is based on the study of a regularized problem. This allows us to exploit the local nature of the singularity. Our results hold with very few smoothness hypotheses on the domain and on the support of the data. We also discuss some extensions of our main results, including the two dimensional case, sources supported on closed curves and on polygonals.

math.AP↗

arXiv2021

Weighted discrete Hardy inequalities on trees and applications

In this paper, we study certain inequalities and a related result for weighted Sobolev spaces on Hölder-$α$ domains, where the weights are powers of the distance to the boundary. We obtain results regarding the divergence equation's solvability, and the improved Poincaré, the fractional Poincaré, and the Korn inequalities. The proofs are based on a local-to-global argument that involves a kind of atomic decomposition of functions and the validity of a weighted discrete Hardy-type inequality on trees. The novelty of our approach lies in the use of this weighted discrete Hardy inequality and a sufficient condition that allows us to study the weights of our interest. As a consequence, the assumptions on the weight exponents that appear in our results are weaker than those in the literature.

math.AP↗

arXiv2018

A weighted setting for the numerical approximation of the Poisson problem with singular sources

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt class. Second, we prove the stability in weighted norms for standard finite element approximations under the quasi-uniformity assumption on the family of meshes.

math.NA↗

arXiv2016

Anisotropic finite elements for elliptic problems with singular data

We study the problem $-\Delta u = \gamma$, where $\gamma$ is a singular measure, with support on a curve or a point. We prove that optimal rates of convergence for the finite element method can be obtained using properly graded meshes. In particular, we consider isotropic graded meshes when $\gamma$ is a point Dirac delta, and anisotropic graded meshes when $\gamma$ is a measure supported on a segment. Numerical experiments are shown that verify our results, and lead to interesting observations.

math.NA↗