Extension Theorems for Sobolev and More General Spaces for Degenerate Open Sets

Abstract

Let Ω be an open set, l ∈ ℕ, 1 ≤ p ≤ ∞. We shall discuss the problem of extension for the Sobolev spaces $W_p^l(\Omega )$ of all functions f ∈ L _p(Ω) for which the weak gradient ${\nabla _l}f = {\left\{ {{D^\alpha }f} \right\}_{|\alpha | = l}}$ exists on Ω and

$$ \parallel f{\parallel _{W_p^l\left( \Omega \right)}} = \parallel f{\parallel _{{L_p}\left( \Omega \right)}} + \parallel {\nabla _t}f{\parallel _{{L_p}\left( \Omega \right)}} < \infty . $$

Here $||{\nabla _l}f|{|_{{L_p}\left( \Omega \right)}} = {\left( {\int_\Omega | {\nabla _l}f{|^p}dx} \right)^{1/p}}and{\nabla _l}f| = {\left( {{\Sigma _{|a| = l}}|{D^a}f{|^2}} \right)^{1/2}}.$

Supported by the Russian Foundation for Basic Research grants 99-01-00843 and 99-01-00868.