scholarly journals Weighted Poincaré Inequalities for Non-local Dirichlet Forms

2015 ◽  
Vol 30 (2) ◽  
pp. 452-489
Author(s):  
Xin Chen ◽  
Jian Wang
Keyword(s):  
Dirichlet Forms ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Non Local ◽  
Weighted Poincaré Inequalities

scholarly journals ORLICZ–POINCARÉ INEQUALITIES

2008 ◽  
Vol 51 (2) ◽  
pp. 529-543 ◽  
Author(s):  
Feng-Yu Wang
Keyword(s):  
Porous Media ◽  
Probability Measure ◽  
Poincaré Inequality ◽  
Dirichlet Forms ◽  
Sobolev Inequalities ◽  
Young Function ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Porous Media Equations

AbstractCorresponding to known results on Orlicz–Sobolev inequalities which are stronger than the Poincaré inequality, this paper studies the weaker Orlicz–Poincaré inequality. More precisely, for any Young function $\varPhi$ whose growth is slower than quadric, the Orlicz–Poincaré inequality$$ \|f\|_\varPhi^2\le C\E(f,f),\qquad\mu(f):=\int f\,\mathrm{d}\mu=0 $$is studied by using the well-developed weak Poincaré inequalities, where $\E$ is a conservative Dirichlet form on $L^2(\mu)$ for some probability measure $\mu$. In particular, criteria and concrete sharp examples of this inequality are presented for $\varPhi(r)=r^p$ $(p\in[1,2))$ and $\varPhi(r)= r^2\log^{-\delta}(\mathrm{e} +r^2)$ $(\delta>0)$. Concentration of measures and analogous results for non-conservative Dirichlet forms are also obtained. As an application, the convergence rate of porous media equations is described.

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.

Some weighted Poincare inequalities

2009 ◽  
Vol 58 (4) ◽  
pp. 1619-1638 ◽  
Author(s):  
Fausto Ferrari ◽  
Enrico Valdinoci
Keyword(s):  
Poincaré Inequalities ◽  
Poincare Inequalities ◽  
Weighted Poincaré Inequalities

scholarly journals Poincaré inequalities and Neumann problems for the variable exponent setting

2021 ◽  
Vol 4 (5) ◽  
pp. 1-22
Author(s):  
David Cruz-Uribe ◽  
◽  
Michael Penrod ◽  
Scott Rodney ◽  
Keyword(s):  
Variable Exponent ◽  
Growth Conditions ◽  
Linear Elliptic Equation ◽  
Poincaré Inequalities ◽  
Neumann Problems ◽  
Variable Exponent Spaces ◽  
Nonstandard Growth ◽  
Poincare Inequalities ◽  
Nonstandard Growth Conditions ◽  
Weighted Poincaré Inequalities

<abstract><p>In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.</p></abstract>

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