A finite element approach to Hölder extension using prefractals (Q390197)

The author considers the problem of extending a given function \(f\) defined on a closed subset \(S \subset \Omega\) to \(\Omega\) so that certain characteristics of the original \(f\) are retained. In particular, a function \(u\) which satisfies the Hölder condition NEWLINE\[NEWLINE |u(x) - u(y)| \leq C_0 |x-y|^\beta, NEWLINE\]NEWLINE for all \(x,y\) on a fractal Koch curve \(S\) is extended to a larger domain \(\Omega\subset\mathbb{R}^2\). It is shown that there exists an extension, \(u^*\), of \(u\) that is defined everywhere on \(\Omega\), is Hölder continuous everywhere in \(\Omega\), and \(u^*|_S \equiv u\). Moreover, \(u^*\) satisfies the estimate NEWLINE\[NEWLINE |u^*|_{\overline{\Omega}, \beta} \leq C\,\|u\|_{S,\beta}, NEWLINE\]NEWLINE where the quantity \(C\) does not depend on \(u\). The construction of \(u^*\) uses the self-similarity properties of the Koch curve \(S\) as well as the iterative process that generates it.