Trace Operator on von Koch’s Snowflake

Abstract

AbstractWe study properties of the boundary trace operator on the Sobolev space $$W^1_1(\Omega )$$ W 1 1 ( Ω ) . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator $$Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)$$ T r : W 1 1 ( Ω K ) → X ( Ω K ) , where $$\Omega _K$$ Ω K is von Koch’s snowflake and $$X(\Omega _K)$$ X ( Ω K ) is a trace space with the quotient norm. Since $$\Omega _K$$ Ω K is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr, i.e. a linear operator $$S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)$$ S : X ( Ω K ) → W 1 1 ( Ω K ) such that $$Tr \circ S= Id_{X(\Omega _K)}$$ T r ∘ S = I d X ( Ω K ) . In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch’s snowflake. Moreover we identify the isomorphism class of the trace space as $$\ell _1$$ ℓ 1 . As an additional consequence of our approach we obtain a simple proof of the Peetre’s theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain $$\Omega $$ Ω with regular boundary, which explains Banach space geometry cause for this phenomenon.