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A new approach to weighted Sobolev spaces

Djamel eddine Kebiche

TL;DR

This work introduces a flexible framework for weighted Sobolev spaces with arbitrarily small weights by replacing the classical distributional derivative with a weak derivative in the sense of $L_{v,\mathrm{loc}}^{1}(\Omega)$. It defines and analyzes the spaces $W_{V,v}^{m,p}(\Omega)$ and $W_{s,v}^{m,p}(\Omega)$, proves completeness and density results for smooth functions, and develops trace and Poincar\'e tools in this weighted setting. The methods are then applied to degenerate elliptic PDEs, establishing existence and uniqueness of weak solutions and revealing scenarios where solutions may fail to be locally integrable. Overall, the paper provides a domain-free, robust approach to managing degenerate weights and broadens the functional analytic toolkit for weighted and non-locally integrable regimes with potential PDE applications.

Abstract

We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré Inequality and a trace operator are developed, and density results of smooth function are established.

A new approach to weighted Sobolev spaces

TL;DR

This work introduces a flexible framework for weighted Sobolev spaces with arbitrarily small weights by replacing the classical distributional derivative with a weak derivative in the sense of . It defines and analyzes the spaces and , proves completeness and density results for smooth functions, and develops trace and Poincar\'e tools in this weighted setting. The methods are then applied to degenerate elliptic PDEs, establishing existence and uniqueness of weak solutions and revealing scenarios where solutions may fail to be locally integrable. Overall, the paper provides a domain-free, robust approach to managing degenerate weights and broadens the functional analytic toolkit for weighted and non-locally integrable regimes with potential PDE applications.

Abstract

We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré Inequality and a trace operator are developed, and density results of smooth function are established.
Paper Structure (9 sections, 17 theorems, 122 equations)

This paper contains 9 sections, 17 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. The weighted Lebesgue space $L_{w}^{p}(\Omega)$
  3. The weak derivative in the sense of $L_{v,\mathrm{loc}}^{1}(\Omega)$
  4. The weighted Sobolev space $W_{V,v}^{m,p}(\Omega)$
  5. Density of smooth functions
  6. The trace operator
  7. Spaces of null trace
  8. Poincaré inequality
  9. Degenerate elliptic PDE

Key Result

Theorem 2

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $w\in E_{p}(\Omega)$, and $p\in[1,\infty)$. Then, the space $L_{w}^{p}(\Omega)$ equipped with the norm (eq:=000020weighted=000020norm) is a Banach space (Hilbert space when $p=2$).

Theorems & Definitions (52)

  • Definition 1
  • Theorem 2
  • Proposition 3
  • proof
  • Definition 4
  • Remark 5
  • Example 6
  • Proposition 7
  • proof
  • Definition 8
  • ...and 42 more