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SIAM J. MATH. ANAL. c 2010 Society for Industrial and Applied Mathematics Vol. 42, No. 4, pp. 14491482
 

Summary: SIAM J. MATH. ANAL. c 2010 Society for Industrial and Applied Mathematics
Vol. 42, No. 4, pp. 1449­1482
TRACE THEOREMS FOR A CLASS OF RAMIFIED DOMAINS
WITH SELF-SIMILAR FRACTAL BOUNDARIES
YVES ACHDOU AND NICOLETTA TCHOU
Abstract. 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

  

Source: Achdou, Yves - Laboratoire Jacques-Louis Lions, Université Pierre-et-Marie-Curie, Paris 6

 

Collections: Mathematics