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Trace operator on von Koch's snowflake

Krystian Kazaniecki, Michał Wojciechowski

Abstract

We study properties of the boundary trace operator on the Sobolev space $W^1_1(Ω)$. Using the density result by Koskela and Zhang, we define a surjective operator \mbox{$Tr: W^1_1(Ω_K)\rightarrow X(Ω_K)$}, where $Ω_K$ is von Koch's snowflake and $X(Ω_K)$ is a trace space with the quotient norm. Since $Ω_K$ is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý that there exists a right inverse to $Tr$, i.e. a linear operator $S: X(Ω_K) \rightarrow W^1_1(Ω_K)$ such that $Tr \circ S= Id_{X(Ω_K)}$. 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 $\ell_1$. As an additional consequence of our approach we obtain a simple proof of the Peetre's theorem about non-existence of the right inverse for domain $Ω$ with regular boundary, which explains Banach space geometry cause for this phenomenon.

Trace operator on von Koch's snowflake

Abstract

We study properties of the boundary trace operator on the Sobolev space . Using the density result by Koskela and Zhang, we define a surjective operator \mbox{}, where is von Koch's snowflake and is a trace space with the quotient norm. Since is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý that there exists a right inverse to , i.e. a linear operator such that . 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 . As an additional consequence of our approach we obtain a simple proof of the Peetre's theorem about non-existence of the right inverse for domain with regular boundary, which explains Banach space geometry cause for this phenomenon.

Paper Structure

This paper contains 3 sections, 14 theorems, 115 equations, 8 figures.

Table of Contents

  1. Properties of $BV(\Omega)$ and trace operator
  2. Proof of Peetre's theorem
  3. Trace operator on von Koch's snowflake

Key Result

Theorem 1

Let $\Omega$ be an open domain with Lipschitz boundary and $\partial\Omega$ be a Jordan curve. Let $Tr:W^{1}_{1}(\Omega )\rightarrow L^1(\partial\Omega)$ be a trace operator. Then there is no continuous, linear operator $S:L^1(\partial\Omega) \rightarrow W^{1}_{1}(\Omega)$ such that $Tr\circ S= Id_

Figures (8)

  • Figure 1: Self similar Whitney decomposition of von Koch's snowflake
  • Figure 2: On the left "pants" shaped polygon and its descendants, on the right "palace" shaped polygon and its descendants
  • Figure 3: One of the six parts of the von Koch's snowflake.
  • Figure 4: First step of the inductive construction of the Whitney Covering.
  • Figure 5: Five regions of the second step of the construction.
  • ...and 3 more figures

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6
  • Definition 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 23 more