Counter-examples of regularity in variable exponent Sobolev spaces. (English) Zbl 1084.46025

Hästö, Peter A.

Poggi-Corradini, Pietro (ed.), The \(p\)-harmonic equation and recent advances in analysis. Proceedings of the 3rd prairie analysis seminar, Manhattan, KS, USA, October 17–18, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3610-2/pbk). Contemporary Mathematics 370, 133-143 (2005).

Summary: An example is presented which demonstrates that continuous functions are not dense in some variable exponent Sobolev spaces. In contrast to previous examples, our example features an exponent which is uniformly continuous and near optimal. Using the same example, we also show that the minimizer of the Dirichlet energy integral is not always continuous and that not quasi-every point of a Sobolev function needs to be a Lebesgue point.
For the entire collection see [Zbl 1061.31001].

MSC:

46E35

Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems