Poincaré and Hardy inequalities

Abstract

Spectral matters form a powerful motivation for the study of these inequalities. Thus let Ω be an open subset of Rn with volume |Ω| and let Δ D,Ω ,Δ N,Ω be respectively the Dirichlet Laplacian and the Neumann Laplacian on Ω. We recall that —Δ D,Ω is the Friedrichs extension of -Δ on C ∞ 0(Ω) and that its domain D(—Δ D,Ω )is contained in the Sobolev space W 1 2(Ω): v is the Dirichlet Laplacian of u ∈ D(—Δ D,Ω ), v = Δu, if it belongs to L 1, 1oc (Ω) and for all φ ∈ C ∞ 0(Ω), $$ \int\limits_\Omega {\nabla u.\nabla \emptyset dx = - \int\limits_\Omega \upsilon } \emptyset dx. $$ KeywordsOrlicz SpaceHardy InequalityContinuous NormBanach Function SpaceYoung FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.