“…The notations in Lemma C.1 are used again here.Setting v − = − min(v, 0) and R λ = u − u + [u λ − u ] − , we have u − u λ = R λ − φ, (φ α , [u λ − u ] − ) = Ω [u λ − u ] α + [u λ − u ] − dP ≡ 0, α > 0. (40) Setting v = u − R λ in(13) and in(17), respectively, we get− ∂u ∂t , −R λ + B(u, −R λ ; t) ≥ (f, −R λ ) and − ∂u λ ∂t , R λ + B(u λ , R λ ; t) + λ(φ 1/k , R λ ) = (f, R λ ), which implies − ∂(u λ − u) ∂t , R λ + B(u λ − u, R λ ; t) + λ(φ 1/k , R λ ) ≥ 0.It follows from u ≤ u and φ ≥ 0 that(φ 1/k , R λ ) = (φ 1/k , u − u ) + (φ 1/k , [u λ − u ] − ) = (φ 1/k , u − u ) ≤ 0.Therefore,− ∂(u − u λ ) ∂t , R λ + B(u − u λ , R λ ; t) ≤ 0,which, together with (40), yields− ∂R λ ∂t , R λ + B(R λ , R λ ; t) ≤ − ∂φ(t) ∂t , R λ + B(φ, R λ ; t).…”