Sobolev embeddings on domains involving two types of symmetries

TL;DR

The paper addresses improving Sobolev embeddings for product domains with cylindrical symmetry by exploiting a compact group $G$ acting on the first factor and radial symmetry on the second. It introduces the $G$-invariant Sobolev space $H^1_{s,G}(\Omega)$ and weighted target spaces $L^q_h(\Omega)$ with $h(y)=\|x_2\|^l$, proving that the embedding into $L^q_h(\Omega)$ is compact for $q$ up to $2^{*}_{k}+\psi$, where $2^{*}_{k}=\frac{2(N-k)}{N-k-2}$ and $\psi$ is explicitly given in two dimensional regimes. The main results are two theorems providing explicit $\psi$ for $\dim\Omega_2=j\ge3$ and for $j=2$, respectively, thereby extending Wang's cylindrical-symmetry results via Hebey–Vaugon-type estimates and Riesz-potential bounds. The approach combines variational techniques, concentration-compactness type arguments, and precise integral estimates to push the range of admissible exponents beyond the usual critical threshold, with implications for PDE existence results under symmetry constraints.

Abstract

It is well known that Sobolev embeddings can be improved in the presence of symmetries. In this article, we considere the situation in which given a domain $Ω=Ω_1 \times Ω_2$ in $\mathbb{R}^N$ with a cylindrical symmetry, and acting a group $G$ in $Ω_1$, for this situation it is shown that the critical Sobolev exponent increases in the case of embeddings into weighted spaces $L^{q}_{h}(Ω)$. In this paper, we will enunciate several results related to compact embeddings of a Sobolev space with radially symmetric functions into some weighted space $L^{q}$, with $q$ higher than the usual critical exponent.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Definition
• Definition
• Lemma 2.1: Wang, 2006
• Lemma 2.2: Wang, 2006
• Definition
• Definition
• Lemma 3.1: Sobolev's lemma
• Definition