Weighted Hardy-Sobolev type inequalities with boundary terms

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.

💡 Research Summary

The paper develops a new class of weighted Hardy‑Sobolev inequalities that incorporate boundary terms and work under very mild regularity assumptions on the domain. The authors consider a domain Ω⊂ℝⁿ described as the region above a continuous graph x_N>ψ(x′) and a non‑negative weight function W(x) that belongs to L¹_loc(Ω) with a weak derivative ∂_{x_N}W∈L¹_loc(Ω) (assumption (W0)).

Two monotonicity regimes for the weight are studied:

1. Increasing weights (W+1). Here W_{x_N}>0 a.e. and W·