Abstract
We study properties of the boundary trace operator on the Sobolev space \(W^1_1(\Omega )\). Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator \(Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)\), where \(\Omega _K\) is von Koch’s snowflake and \(X(\Omega _K)\) is a trace space with the quotient norm. Since \(\Omega _K\) 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 \(S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)\) such that \(Tr \circ S= Id_{X(\Omega _K)}\). 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 (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain \(\Omega \) with regular boundary, which explains Banach space geometry cause for this phenomenon.
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Acknowledgements
We would like to thank Anna Kamont for valuable comments and suggestions.
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Open access funding provided by Johannes Kepler University Linz. This research was partially supported by the National Science Centre, Poland, and Austrian Science Foundation FWF joint CEUS programme. National Science Centre project no. 2020/02/Y/ST1/00072 and FWF project no. I5231.
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K.K. and M.W. wrote the main manuscript text and K.K. prepared figures. All authors reviewed the manuscript.
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Appendix
Appendix
In this section we show the steps of the construction of the Whitney Covering of the von Koch’s snowflake used in our proof. We divide the von Koch’s snowflake into six identical parts. We focus our attention on one of them.
We start our construction with the third step of the iterative construction of von Koch’s snowflake. In most steps of the construction we limit the description of it to showing pictures. In our construction all the lines we use are parallel to the sides of the equilateral triangle (the starting point for von Koch’s snowflake construction). We denote the vectors pointing in those three directions respectively by \(\nu _1,\nu _2,\nu _3\) and by L(X, v) we denote the line parrallel to a vector v, which contains point X. In the first step we construct vertices of the first generation of polygons - “pants” polygon.
The green points are intersections of following lines:
Now we take next generation of the approximation of von Koch’s snowflake. We observe that we can cover the boundary using few blue and lime regions.
In blue regions we repeat the first step of the construction. In the lime part we do the following (Fig. 6).
and \(R(6)=P(2)\), \(R(7)=P(1)\), where P(1), P(2) are points from the previous step of construction. In the end we have constructed five new polygons. What is important vertices on the boundary of given region coincide with the corresponding vertices from the neighboring regions (Fig. 7).
We take the next iterative step of the approximation of von Koch’s snowflake and we observe that again we can cover the neighborhood of the boundary by the lime and blue regions.
In every region we repeat the construction according to the color of the region. Let us observe that in the next generation every lime region has 3 subregions (lime,blue,lime) and every blue region has 5 subregions (lime,blue,blue,blue,lime). Since the vertices of neighboring polygons coincide we can repeat the construction inductively.
Let \(K_n\) be n-th approximation of von Koch’s snowflake. By \(G_n\) we denote the set covered by polygons from n-th generation of the construction, where \(G_0\) is just the six pointed star in middle of von Koch’s snowflake.
The polygons on the n-th step of the construction almost cover the set \(K_{n+2}\) (except a narrow strip next to the boundary). Observe that we always perform the same construction on lime and blue regions. However the regions on n-th step are the scaled copies of regions from the second step with a scale \(\frac{1}{3^{n-2}}\) for \(n\geqslant 2\). Therefore there exists a constant \(C>0\) such that
Since for \(k\in \mathbb {N}\) we have \(K_n\subset K_{n+k}\) and for any \(x\in K_n\) the sequence \({\text {dist}}(x,\partial K_{n+j})\) is non-increasing. We get
Therefore
where \(\Omega _K\) is von Koch’s snowflake. Obviously
Hence the family of polygons we constructed covers von Koch’s snowflake. Other properties of the Whitney covering follow easily from the construction.
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Kazaniecki, K., Wojciechowski, M. Trace Operator on von Koch’s Snowflake. Potential Anal (2024). https://doi.org/10.1007/s11118-024-10124-w
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DOI: https://doi.org/10.1007/s11118-024-10124-w