On an Extension of the First Korn Inequality to Incompatible Tensor Fields on Domains of Arbitrary Dimensions

For a bounded domain \(\varOmega \) in \(\mathbb {R}^N\) with Lipschitz boundary \(\varGamma =\partial \varOmega \) and a relatively open and non-empty subset \(\varGamma _t\) of \(\varGamma \), we prove the existence of a positive constant \(c\) such that inequality \( c\Vert T\Vert _{\mathsf {L}^2(\varOmega ,\mathbb {R}^{N\times N})} \le \Vert {{\mathrm{sym}}}T\Vert _{\mathsf {L}^2(\varOmega ,\mathbb {R}^{N\times N})} +\Vert {{\mathrm{Curl}}}T\Vert _{\mathsf {L}^2(\varOmega ,\mathbb {R}^{N\times N(N-1)/2})}\) holds for all tensor fields \(T \in \overset{\circ }{\mathsf {H}}({{\mathrm{Curl}}};\varGamma _t,\varOmega ,\mathbb {R}^{N\times N})\), this is, for all \(T:\varOmega \rightarrow \mathbb {R}^{N\times N}\) which are square-integrable and possess a row-wise square-integrable rotation tensor field \({{\mathrm{Curl}}}T:\varOmega \rightarrow \mathbb {R}^{N\times N(N-1)/2}\) and vanishing row-wise tangential trace on \(\varGamma _t\). For compatible tensor fields \(T=\nabla { v}\) with \({ v}\in \mathsf {H}^1(\varOmega ,\mathbb {R}^N)\) having vanishing tangential Neumann trace on \(\varGamma _t\) the inequality reduces to a non-standard variant of the first Korn inequality since \({{\mathrm{Curl}}}T=0\), while for skew-symmetric tensor fields \(T\) the Poincare inequality is recovered. If \(\varGamma _t=\emptyset \), our estimate still holds at least for simply connected \(\varOmega \) and for all tensor fields \(T \in \mathsf {H}({{\mathrm{Curl}}};\varOmega ,\mathbb {R}^{N\times N})\) which are \(\mathsf {L}^2(\varOmega ,\mathbb {R}^{N\times N})\)-perpendicular to \({{\mathrm{\mathfrak {so}}}}(N)\), i.e., to all skew-symmetric constant tensors.