Classical regularity theory for linear problems

Abstract

In this chapter we begin studying the local behavior of (weak) solutions$u \in H_{{\text{loc}}}^1\left( {\Omega ;{\mathbb{R}^m}} \right)$ of a system of equations given by

$$\begin{array}{*{20}{c}} { - \sum\limits_{\alpha ,\beta ,j} {{\partial _{{\chi _\alpha }}}\left( {A_{ij}^{\alpha \beta }{\partial _{{\chi _\beta }}}{u^j}} \right) = {f_i} - \sum\limits_\alpha {{\partial _{{\chi _\alpha }}}F_i^\alpha } } }&{i = 1, \ldots ,m} \end{array}$$

((2.1))

with$A_{ij}^{\alpha \beta } \in {L^\infty }(\Omega ;\mathbb{R}),\,{f_i} \in L_{{\text{loc}}}^2(\Omega ;\mathbb{R})$ and$F_i^\alpha \in L_{{\text{loc}}}^2(\Omega ;\mathbb{R})$.