Trace Theorems for a Class of Ramified Domains with Self-Similar Fractal Boundaries

This work deals with trace theorems for a class of ramified bidimensional domains with a self-similar fractal boundary . The fractal boundary is supplied with a probability measure called the self-similar measure. Emphasis is put on the case when the domain is not an domain as defined by Jones and the fractal set is not totally disconnected. In this case, the classical trace results cannot be used. Here, the Lipschitz spaces with jumps recently introduced by Jonsson play a crucial role. Indeed, it is proved in particular that if the Hausdorff dimension d of is not smaller than one, then the space of the traces of functions in W m+1,q(), m N, 1 < q < , is JLip(m + 1 - 2-d q , q, q; m; ). The proof is elementary; a main step is a strengthened trace inequality in the norm Lq (). Key words. function spaces, trace theorem, fractal boundary AMS subject classifications. 46E35, 28A80, 42C40 DOI. 10.1137/090747294

• Type: Journal
• Year: 2010
• Where: SIAMMA
• Authors: Yves Achdou, Nicoletta Tchou
• Keywords: Fractal Boundary, Self-similar Fractal Boundary, SIAMMA 2010, Trace Theorem