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Weighted Hardy-Sobolev type inequalities with boundary terms

João Marcos do Ò, Marcelo Furtado, Everaldo Medeiros, Jesse Ratzkin

TL;DR

This work develops a broad framework of weighted Hardy–Sobolev inequalities on domains lying above a graph under mild monotonicity assumptions on a weight $W$, yielding Type I (increasing weight) and Type II (decreasing weight) inequalities with explicit boundary terms. By coupling these inequalities with classical Sobolev and Trace embeddings, the authors derive continuous weighted embeddings for the associated function spaces $\mathcal{D}_W^+$ and $\mathcal{D}_W^-$, including subcritical, critical, and trace results, valid even on unbounded domains with non-smooth boundary graphs. The results generalize and sharpen several known inequalities (e.g., non-singular weight, boundary-trace inequalities) and provide concrete weight examples illustrating sharpness and applicability to elliptic problems with Neumann or Robin boundary conditions. The paper also sketches variational applications to nonlinear PDEs, highlighting how the weighted inequalities inform energy estimates and the existence of solutions, and outlines several directions for further research, including optimal constants and extensions to higher co-dimension boundaries. Overall, the work advances the theory of weighted functional inequalities on irregular unbounded domains and broadens the toolkit for studying elliptic problems with boundary interactions in complex geometries.

Abstract

In this paper we establish a new class of weighted Hardy-Sobolev type inequalities under mild monotonicity assumptions on the weight function. As a consequence, we derive the corresponding weighted Sobolev and trace-type inequalities. These results play an important role in the analysis of elliptic problems with Neumann or Robin boundary conditions in unbounded domains.

Weighted Hardy-Sobolev type inequalities with boundary terms

TL;DR

This work develops a broad framework of weighted Hardy–Sobolev inequalities on domains lying above a graph under mild monotonicity assumptions on a weight , yielding Type I (increasing weight) and Type II (decreasing weight) inequalities with explicit boundary terms. By coupling these inequalities with classical Sobolev and Trace embeddings, the authors derive continuous weighted embeddings for the associated function spaces and , including subcritical, critical, and trace results, valid even on unbounded domains with non-smooth boundary graphs. The results generalize and sharpen several known inequalities (e.g., non-singular weight, boundary-trace inequalities) and provide concrete weight examples illustrating sharpness and applicability to elliptic problems with Neumann or Robin boundary conditions. The paper also sketches variational applications to nonlinear PDEs, highlighting how the weighted inequalities inform energy estimates and the existence of solutions, and outlines several directions for further research, including optimal constants and extensions to higher co-dimension boundaries. Overall, the work advances the theory of weighted functional inequalities on irregular unbounded domains and broadens the toolkit for studying elliptic problems with boundary interactions in complex geometries.

Abstract

In this paper we establish a new class of weighted Hardy-Sobolev type inequalities under mild monotonicity assumptions on the weight function. As a consequence, we derive the corresponding weighted Sobolev and trace-type inequalities. These results play an important role in the analysis of elliptic problems with Neumann or Robin boundary conditions in unbounded domains.
Paper Structure (15 sections, 11 theorems, 129 equations)

This paper contains 15 sections, 11 theorems, 129 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Weighted Hardy inequality and consequences: increasing weight case
  3. Weighted Hardy inequality and consequences: decreasing weight case
  4. Comparison with previous results
  5. Organization of the paper
  6. Some examples of weights
  7. Increasing weights
  8. Decreasing weights
  9. Proofs and Consequences in the Increasing Case
  10. Weighted Sobolev embeddings: the increasing case
  11. Proofs and Consequences in the decreasing Case
  12. Weighted Sobolev embeddings: The decreasing case
  13. Final comments
  14. Applications to nonlinear PDE's
  15. Further Research Directions

Key Result

Theorem 1.1

Suppose W_0 and W_1^+. If $p>1$ then, for any $u \in C_0^\infty(\mathbb{R}^N)$, the following inequality is satisfied:

Theorems & Definitions (37)

  • Theorem 1.1: Hardy inequality of type I
  • Theorem 1.2: Hardy inequality of type II
  • Remark 1.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • ...and 27 more