  In this article we study basic properties of the mixed BV-Sobolev capacity
with variable exponent p. We give an alternative way to define mixed type
BV-Sobolev-space which was originally introduced by Harjulehto, H\"ast\"o, and
Latvala. Our definition is based on relaxing the p-energy functional with
respect to the Lebesgue space topology. We prove that this procedure produces a
Banach space that coincides with the space defined by Harjulehto et al. for
bounded domain and log-H\"older continuous exponent p. Then we show that this
induces a type of variable exponent BV-capacity and that this is a Choquet
capacity with many usual properties. Finally, we prove that this capacity has
the same null sets as the variable exponent Sobolev capacity when p is
log-H\"older continuous.
