Global Compensated Compactness Theorem for General Differential Operators of First Order

Let A₁(x, D) and A₂(x, D) be differential operators of the first order acting on l-vector functions u = (u¹, . . . , u¹) in a bounded domain Ω ⊂ ℝⁿ with the smooth boundary ∂Ω. We assume that the H¹-norm, is equivalent to, where B_i = B_i(x, ν) is the trace operator onto ∂ Ω associated with A_i(x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ∂Ω. Furthermore, we impose on A₁ and A₂ a cancellation property such as A₁A₂′ = 0 and A₂A₁′ = 0, where A_i′ is the formal adjoint differential operator of A_i(i = 1, 2). Suppose that and converge to u and v weakly in L²(Ω), respectively. Assume also that and are bounded in L²(Ω). If either or is bounded in H^1/2(∂Ω), then it holds that. We also discuss a corresponding result on compact Riemannian manifolds with boundary.