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

  Published:2018-03-27
 Printed: Dec 2018
  • 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 degenerate Sobolev space, $p$-Laplacian, Poincaré inequalities
MSC Classifications: 30C65, 35B65, 35J70, 42B35, 42B37, 46E35 show english descriptions Quasiconformal mappings in ${\bf R}^n$, other generalizations
Smoothness and regularity of solutions
Degenerate elliptic equations
Function spaces arising in harmonic analysis
Harmonic analysis and PDE [See also 35-XX]
Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
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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