arXiv subjects

Joaquín Domínguez-de-Tena

Publications and source records attributed to Joaquín Domínguez-de-Tena.

4 recordsLinked to original sources

arXiv2026

A brush problem. Homogenization involving thin domains and PDEs in graphs

This work analyses the homogenization of a linear elliptic equation with Neumann boundary conditions in a comb/brush domain, composed of a fixed base and a family of thin teeth. The teeth are defined as rescalings of order less than or equal to $\varepsilon$ of a model tooth of arbitrary shape. Periodicity in their distribution is not assumed; instead, the existence of an asymptotic limit density $θ$, which may vanish in certain regions, is assumed. The convergence analysis is performed using an adaptation of the unfolding operator method to a non-periodic framework. Finally, it is shown that, under certain conditions on the geometry of the teeth, the resulting limit problem can be interpreted as a differential equation on a graph.

math.AP↗

arXiv2024

Asymptotic behaviour of the heat equation in an exterior domain with general boundary conditions II. The case of bounded and of $L^{p}$ data

In this work, we study the asymptotic behaviour of solutions to the heat equation in exterior domains, i.e., domains which are the complement of a smooth compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann ones. In this second part of our work, we consider the case of bounded initial data and prove that, after some correction term, the solutions become close to the solutions in the whole space and show how complex behaviours appear. We also analyse the case of initial data in $L^p$ with $1<p<\infty$ where all solutions essentially decay to $0$ and the convergence rate could be arbitrarily slow.

math.AP↗

arXiv2024

Asymptotic behaviour of the heat equation in an exterior domain with general boundary conditions I. The case of integrable data

In this work, we study the asymptotic behaviour of solutions to the heat equation in exterior domains, i.e., domains which are the complement of a smooth compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann conditions for integrable initial data in $L^1(Ω)$. After taking into account the loss of mass of the solution through the boundary, depending on the boundary conditions, we describe the asymptotic spatial distribution of the remaining mass in terms of the Gaussian and of a suitable asymptotic profile function. We show that our results have optimal time rates.

math.AP↗

arXiv2023

On the loss of mass for the heat equation in an exterior domain with general boundary conditions

In this work, we study the decay of mass for solutions to the heat equation in exterior domains, i.e., domains which are the complement of a compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann conditions. We determine the exact amount of mass loss and identify criteria for complete mass decay, in which the dimension of the space plays a key role. Furthermore, the paper provides explicit mass decay rates.

math.AP↗