tailieunhanh - Báo cáo hóa học: " Research Article Hölder Quasicontinuity in Variable Exponent Sobolev Spaces"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Hölder Quasicontinuity in Variable Exponent Sobolev Spaces | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 32324 18 pages doi 2007 32324 Research Article Holder Quasicontinuity in Variable Exponent Sobolev Spaces Petteri Harjulehto Juha Kinnunen and Katja Tuhkanen Received 28 May 2006 Revised 6 November 2006 Accepted 25 December 2006 Recommended by H. Bevan Thompson We show that a function in the variable exponent Sobolev spaces coincides with a Holder continuous Sobolev function outside a small exceptional set. This gives us a method to approximate a Sobolev function with Holder continuous functions in the Sobolev norm. Our argument is based on a Whitney-type extension and maximal function estimates. The size of the exceptional set is estimated in terms of Lebesgue measure and a capacity. In these estimates we use the fractional maximal function as a test function for the capacity. Copyright 2007 Petteri Harjulehto et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Our main objective is to study the pointwise behaviour and Lusin-type approximation of functions which belong to a variable exponent Sobolev space. In particular we are interested in the first-order Sobolev spaces. The standard Sobolev space w1 p R with 1 p 00 consists of functions u e Lp R whose distributional gradient Du D1u . D U also belongs to Lp R . The rough philosophy behind the variable exponent Sobolev space w1 p R is that the standard Lebesgue norm is replaced with the quantity u x I p x dx R where p is a function of x. The exact definition is presented below see also 1 2 . Variable exponent Sobolev spaces have been used in the modeling of electrorheological fluids see for example 3-7 and references therein. Very recently Chen et al. have introduced a new variable exponent model for image restoration 8 . A .

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