Global Compensated Compactness Theorem for General Differential Operators of First Order

Abstract

Let A1(x, D) and A2(x, D) be differential operators of the first order acting on l-vector functions \({u= (u_1, \ldots, u_l)}\) in a bounded domain \({\Omega \subset \mathbb{R}^{n}}\) with the smooth boundary \({\partial\Omega}\). We assume that the H1-norm \({\|u\|_{H^{1}(\Omega)}}\) is equivalent to \({\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_1u\|_{H^{\frac{1}{2}}(\partial\Omega)}}\) and \({\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_2u\|_{H^{\frac{1}{2}}(\partial\Omega)}}\), where Bi = Bi(x, ν) is the trace operator onto \({\partial\Omega}\) associated with Ai(x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to \({\partial\Omega}\)). Furthermore, we impose on A1 and A2 a cancellation property such as \({A_1A_2^{\prime}=0}\) and \({A_2A_1^{\prime}=0}\), where \({A^{\prime}_i}\) is the formal adjoint differential operator of Ai(i = 1, 2). Suppose that \({\{u_m\}_{m=1}^{\infty}}\) and \({\{v_m\}_{m=1}^{\infty}}\) converge to u and v weakly in \({L^2(\Omega)}\), respectively. Assume also that \({\{A_{1}u_m\}_{m=1}^{\infty}}\) and \({\{A_{2}v_{m}\}_{m=1}^{\infty}}\) are bounded in \({L^{2}(\Omega)}\). If either \({\{B_{1}u_m\}_{m=1}^{\infty}}\) or \({\{B_{2}v_m\}_{m=1}^{\infty}}\) is bounded in \({H^{\frac{1}{2}}(\partial\Omega)}\), then it holds that \({\int_{\Omega}u_m\cdot v_m \,{\rm d}x \to \int_{\Omega}u\cdot v \,{\rm d}x}\). We also discuss a corresponding result on compact Riemannian manifolds with boundary.