Wolf, Jög
A Generalization of the Fundamental Estimates for W^m,p - Solutions of Linear Systems with Constant Coefficients (the case 1 < p < 2)
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

35D10 Regularity of generalized solutions

35E99 None of the above but in this section

Abstract: The aim of the present paper is to extent the well known fundamental estimates (w.r.t. the L²-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.
Keywords: elliptic systems, multiplicative inequality, fundamental estimate