Boundary Gradient Estimates

Abstract

An examination of the proof of Theorem 11.5 shows that for elliptic operators of the forms (11.7) or (11.8) the solvability of the classical Dirichlet problem with smooth data depends only upon the fulfillment of Step II of the existence procedure, that is, upon the existence of a boundary gradient estimate. In this chapter we provide a variety of hypotheses for the general equation,

$$Qu = {a^{ij}}\left( {x,u,Du} \right){D_{ij}}u + b\left( {x,u,Du} \right) = 0$$

(14.1)

in Ω ⊂ ℝⁿ, that guarantee a boundary gradient estimate for solutions. These hypotheses are combinations of structural conditions on Q and geometric conditions on the domain Ω. It will be seen that the gradient bound aspect of the theory of quasilinear elliptic equations is not as profound as other aspects such as the Holder estimates of Chapters 6 and 13. The boundary gradient estimates are tied through the classical maximum principle to judicious and generally natural choices of barrier functions. Nevertheless these estimates are of considerable importance since they seem to be the principal factor in determining the solvability character of the Dirichlet problem. This will be evidenced by the non-existence results at the end of the chapter.