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Asymptotic properties of PDEs in compact spaces

Lucía López-Somoza, F. Adrián F. Tojo

TL;DR

The paper addresses existence and asymptotic analysis of PDE solutions on unbounded domains by compactifying the domain via a map $\kappa$ and developing the function space ${\mathcal{C}}^m_{\kappa,\varphi}(\overline A)$. It proves an ${\rm Ascoli{-}Arzelà}$-type compactness result and builds a fixed-point index framework on cones to obtain existence and multiplicity results for integral reformulations of PDEs. A key contribution is the equivalence between relative compactness of a family in ${\mathcal{C}}^m_{\kappa,\varphi}(\overline A)$ and the relative compactness of all associated $\Gamma_p$-images in ${\mathcal{C}}(X,\mathbb{R})$, enabling practical compactness checks. The theory is validated via a concrete hyperbolic PDE example with prescribed asymptotics, demonstrating the utility of domain compactification for PDE analysis.

Abstract

In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of Ascoli-Arzelà Theorem, develop the fixed point index results necessary to prove existence and multiplicity of solutions in these spaces and also illustrate the applicability of the theory with an example.

Asymptotic properties of PDEs in compact spaces

TL;DR

The paper addresses existence and asymptotic analysis of PDE solutions on unbounded domains by compactifying the domain via a map and developing the function space . It proves an -type compactness result and builds a fixed-point index framework on cones to obtain existence and multiplicity results for integral reformulations of PDEs. A key contribution is the equivalence between relative compactness of a family in and the relative compactness of all associated -images in , enabling practical compactness checks. The theory is validated via a concrete hyperbolic PDE example with prescribed asymptotics, demonstrating the utility of domain compactification for PDE analysis.

Abstract

In this article we combine the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. We define new spaces of functions on which study the equations, prove a version of Ascoli-Arzelà Theorem, develop the fixed point index results necessary to prove existence and multiplicity of solutions in these spaces and also illustrate the applicability of the theory with an example.
Paper Structure (4 sections, 7 theorems, 41 equations, 1 figure)

This paper contains 4 sections, 7 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries: compactifications and extensions
  3. The space of continuously $\boldmath{n}$-differentiable $\boldmath{ (\kappa,\varphi)}$-extensions
  4. Fixed points of integral equations

Key Result

Proposition 2.9

Let $X$, $Y$ and $Z$ be metric spaces, $Y$ compact, $\kappa:X\to Y$ a compactification of $X$, $y\in Y\backslash \kappa(X)$ and $f:X\to Z$. Then $\lim_{x\to y}^\kappa f(x)=z$ if and only if for every $\varepsilon\in{\mathbb R}^+$ there exists $\delta\in{\mathbb R}^+$ such that $d(f(x),z)<\varepsilon

Figures (1)

  • Figure 3.1: Representation of the function $f$ (left) and its derivative (right).

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3: Directional compactification
  • Example 2.4: Projective spaces
  • Example 2.5: Alexandroff's one-point compactification
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Proposition 2.9
  • Proposition 2.10: chandler
  • ...and 18 more