Weak Solutions and a Priori Estimates

In this chapter we place no restriction on the values of p >1. The theory is markedly different in the degenerate and singular cases and we will later restrict our attention to p >2. The results extend to a variety of equations and, in particular, to equations with general principal parts satisfying appropriate structure assumptions and with lower order terms. We have chosen to present the results and the proofs for the particular model case (2.1) to bring to light what we feel are the essential features of the theory. Remarks on generalizations, which in some way or another correspond to more or less sophisticated technical improvements, are left to a later section.