L²-regularity: The Caccioppoli inequality

Abstract

In this chapter we discuss regularity in terms of square summability of the derivatives of weak solutions to a linear elliptic system

$$ - D_\alpha (A_{ij}^{\alpha \beta } D_\beta u^j ) = f_i - D_\alpha f_i^\alpha $$

in dependence of the regularity of the coefficients and boundary data, i.e., we deal with the energy estimates for the derivatives of solutions. The basic tool we use is the Caccioppoli inequality, sometimes also called reverse Poincaré inequality, which enables us to give a priori estimates of the L²-norm of the derivatives of a solution u in terms of the L²-norm of u.