A Generalization of the Fundamental Estimates for Wm,p- Solutions of Linear Systems with Constant Coefficients (the case 1 < p < 2)

The aim of the present paper is to extent the well known fundamental estimates (w.r.t. the $L^2$-norm) for weak solutions of a linear elliptic system with constant coefficients: \[ \sum_{ j= 1}^ N \sum_{\mid \alpha\mid,\mid \beta\mid = m} D^\alpha (A_{ij}^{\alpha\beta} D^\beta u^j)=0 \quad \mbox{in}\;\; \Omega\quad(i=1,\ldots,N), \] where $\nu_\circ\!\parallel\!\! \xi\!\!\parallel^2 \leq A_{ij}^{\alpha\beta}\xi_\alpha^i \xi_\beta^j \leq c_\circ \parallel\!\! \xi\!\!\parallel^2\;\forall\xi\in \R^{nN}, (\Omega\subset \R^n$ is open and bounded). Based on a generalization of the "CACCOIOPPOLI - inequality" we are able to establish the extended fundamental estimates w.r.t. the $L^p$- norm of $W^m,p$- solutions (1 < p < 2) of the linear system