Continuity of weakly monotone Sobolev functions of variable exponent. (English) Zbl 1119.31002

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 127-143 (2006).
Sobolev spaces \(W^{1,q}(D),\) where \(D \subset \mathbb R^n\) is an open set, have been extensively studied by the second author from the point of view of boundary behavior. The authors consider the natural question of extending these results to the case of so called variable exponent Sobolev spaces \(W^{1,p(\cdot)}(D)\) where the function \(p:D \to (0,\infty)\) satisfies certain continuity hypotheses. These spaces have been the subject of many recent papers e.g. by D. E. Edmunds and J. Rakosnik [Math. Nachr. 246–247, 53–67 (2002; Zbl 1030.46033)], L. Diening [Math. Nachr. 268, 31–43 (2004; Zbl 1065.46024)], M. Ruzicka [Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, 1748. Springer-Verlag, Berlin (2000; Zbl 0962.76001)], P. Harjulehto, P. Hästö and V. Latvala [Math. Z. 254, No. 3, 591–609 (2006; Zbl 1109.46037)].
The authors prove several interesting results. For instance they establish the continuity of weakly monotone functions in \(W^{1,p(\cdot)}(D)\) under a condition on \(p(\cdot)\) and \(0\)-Hölder-continuity of continuous Sobolev functions under various conditions on the parameters. They also show the sharpness of the results. Finally they investigate the existence of tangential boundary limits of weakly monotone functions of the unit ball \(B \subset \mathbb R^n\) off an exceptional set \(E \subset \partial B\) of vanishing \(p(\cdot)\)-capacity.
For the entire collection see [Zbl 1102.31001].


31B15 Potentials and capacities, extremal length and related notions in higher dimensions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems