Capacity Theory for Monotone Operators

If Au = - div(a(x, Du)) is a monotone operator defined on the Sobolev space W^1,P(Rⁿ, 1 < p < + , with a(x,0) = 0 for a.e. x Rⁿ, the capacity C_A(E,F) relative to A can be defined for every pair (E,F) of bounded sets in Rⁿ with E F. The main properties of the set function C_A(E,F) are investigated. In particular it is proved that C_A(E,F) is increasing and countably subadditive with respect to E, decreasing with respect to F, and continuous, in a suitable sense, with respect to E and F.