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# Poincaré Inequalities and Neumann Problems for the $p$-Laplacian

Published:2018-03-27

• David Cruz-Uribe,
Department of Mathematics, University of Alabama , Tuscaloosa, Alabama 35487, USA
• Scott Rodney,
Dept. of Mathematics, Physics and Geology, Cape Breton University, Sydney, Nova Scotia B1Y3V3
• Emily Rosta,
Dept. of Mathematics, Physics and Geology, Cape Breton University, Sydney, Nova Scotia B1Y3V3
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## Abstract

We 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.
 Keywords: degenerate Sobolev space, $p$-Laplacian, Poincaré inequalities
 MSC Classifications: 30C65 - Quasiconformal mappings in ${\bf R}^n$, other generalizations 35B65 - Smoothness and regularity of solutions 35J70 - Degenerate elliptic equations 42B35 - Function spaces arising in harmonic analysis 42B37 - Harmonic analysis and PDE [See also 35-XX] 46E35 - Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems

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