scholarly journals A remark on optimal weighted Poincaré inequalities for convex domains

2012 ◽  
Vol 23 (4) ◽  
pp. 467-475 ◽  
Author(s):  
Vincenzo Ferone ◽  
Carlo Nitsch ◽  
Cristina Trombetti
Keyword(s):  
Convex Domains ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Weighted Poincaré Inequalities

scholarly journals Poincaré Inequalities and Neumann Problems for the p-Laplacian

2018 ◽  
Vol 61 (4) ◽  
pp. 738-753 ◽  
Author(s):  
David Cruz-Uribe ◽  
Scott Rodney ◽  
Emily Rosta
Keyword(s):  
Quadratic Form ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Poincaré Inequalities ◽  
Existence Of Weak Solutions ◽  
Neumann Problems ◽  
Poincare Inequalities ◽  
Associated Matrix ◽  
Weighted Poincaré Inequalities

AbstractWe prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.

ESTIMATES OF BEST CONSTANTS FOR WEIGHTED POINCARÉ INEQUALITIES ON CONVEX DOMAINS

2006 ◽  
Vol 93 (1) ◽  
pp. 197-226 ◽  
Author(s):  
SENG-KEE CHUA ◽  
RICHARD L. WHEEDEN
Keyword(s):  
Poincaré Inequality ◽  
Concave Function ◽  
Continuous Functions ◽  
Directional Derivatives ◽  
Convex Domains ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Best Constant ◽  
Lipschitz Continuous ◽  
Poincare Inequalities

Let $1 \le q \le p <\infty$ and let $\mathcal{C}$ be the class of all bounded convex domains $\Omega$ in $\mathbb{R}^n$. Following the approach in `An optimal Poincaré inequality in $L^1$ for convex domains', by G. Acosta and R. G. Durán (Proc. Amer. Math. Soc. 132 (2003) 195–202), we show that the best constant $C$ in the weighted Poincaré inequality$$ \| f - f_{av} \|_{L^q_w (\Omega)} \le C w(\Omega)^{\frac{1}{q} - \frac{1}{p}} \mbox{diam}(\Omega) \| \nabla f \|_{L^p_w(\Omega)} $$for all $\Omega \in \mathcal{C}$, all Lipschitz continuous functions $f$ on $\Omega$, and all weights $w$ which are any positive power of a non-negative concave function on $\Omega$ is the same as the best constant for the corresponding one-dimensional situation, where $\mathcal{C}$ reduces to the class of bounded intervals. Using facts from `Sharp conditions for weighted 1-dimensional Poincaré inequalities', by S.-K. Chua and R. L. Wheeden (Indiana Math. J. 49 (2000) 143–175), we estimate the best constant. In the case $q = 1$ and $1 <\infty$, our estimate is between the best constant and twice the best constant. Furthermore, when $p = q = 1$ or $p = q = 2$, the estimate is sharp. Finally, in the case where the domains in $\mathbb{R}^n$ are further restricted to be parallelepipeds, we obtain a slightly different form of Poincaré's inequality which is better adapted to directional derivatives and the sidelengths of the parallelepipeds. We also show that this estimate is sharp for a fixed rectangle.

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