Admissible function spaces for weighted Sobolev inequalities

T. V. Anoop; Nirjan Biswas; Ujjal Das

doi:10.3934/cpaa.2021105

Article Contents

2021, Volume 20, Issue 9: 3259-3297. Doi: 10.3934/cpaa.2021105

This issue Previous Article Delta waves and vacuum states in the vanishing pressure limit of Riemann solutions to Baer-Nunziato two-phase flow model Next Article

Admissible function spaces for weighted Sobolev inequalities

1.
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
2.
The Institute of Mathematical Sciences (HBNI), Chennai 600113, India

^* Corresponding author
^* Corresponding author

Received: December 2020

Revised: May 2021

Early access: June 2021

Published: September 2021

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Abstract

Let $ k,N\in \mathbb{N} $ with $ 1\le k\le N $ and let $ \Omega = \Omega_1 \times \Omega_2 $ be an open set in $ \mathbb{R}^k \times \mathbb{R}^{N-k} $. For $ p\in (1,\infty) $ and $ q \in (0,\infty), $ we consider the following weighted Sobolev type inequality:

$\begin{align} \int_{\Omega} |g_1(y)||g_2(z)| |u(y,z)|^q \, {\rm d}y {\rm d}z \leq C \left( \int_{\Omega} | \nabla u(y,z) |^p \, {\rm d}y {\rm d}z \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(\Omega), \\(0.1)\end{align}$

for some $ C>0 $. Depending on the values of $ N,k,p,q $ we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for $ (g_1, g_2) $ so that (0.1) holds. Furthermore, we give a sufficient condition on $ g_1,g_2 $ so that the best constant in (0.1) is attained in the Beppo-Levi space $ \mathcal{D}^{1,p}_0(\Omega) $-the completion of $ \mathcal{C}^1_c(\Omega) $ with respect to $\|\nabla u\|_{L p(\Omega)}$.

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Mathematics Subject Classification: Primary: 35A23, 46E30, 46E35, 47J30.

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