On the operator of symmetric differentiation on a compact Riemannian manifold with boundary

Abstract

We state a particular case of one of the theorems which we shall prove. Let Ω be a bounded open set in ℝnwith smooth boundary and let σ=(σij)be a symmetric second-order tensor with components σijeHk(Ω) for some (positive or negative) integer k; Hkare Sobolev spaces on Ω. Then we have \(\sigma _{ij} = \frac{{\partial u_i }}{{\partial x_j }} + \frac{{\partial u_j }}{{\partial x_i }}\) for some uieHk+1(Ω),i=1,...,n, if and only if\(\mathop \Sigma \limits_{i,j} \mathop \smallint \limits_\Omega \sigma _{ij} \omega ^{ij} dx = 0\) (if k<0, the integral is in fact a duality) for any symmetric tensor (ω with components \(\omega ^{ij} \in H_0^\infty (\Omega ) = \mathop \cap \limits_{m \geqq 0} H_0^m (\Omega )\) and such that\(\mathop \Sigma \limits_j \frac{{\partial \omega ^{ij} }}{{\partial x_j }} = 0\)). Some applications in the theory of elasticity are also given.