Apriori Bounds

Abstract

This chapter is mostly preparation for the next chapter in which existence theorems for the oblique boundary derivative problem $\mathbf {L} u = f$ on $\Omega$ and $\mathbf {M}u = g$ on $\partial \Omega$ are proven. This problem is first solved for the case $\triangle u = f$ on $R^n_+$ and $\mathbf {M}_0u = 0$ on $R^n_0$, where $\mathbf {M}_0$ is a boundary derivative operator with constant coefficients, by constructing a Green function $G_{\frac{1}{2}}$ for the half space $R^n_+$ incorporating the boundary condition. This step is followed by deriving bounds on weighted Hölder norms $|u;\rho |_{2+\alpha }^{(b)}$ assuming that $u$ is a solution of the problem $\triangle u = f$ on $B^+$. Such bounds are called “aprior” as it is assumed that such a solution exists. These results are then extended to operators $\mathbf {L}_0$ and $\mathbf {M}_0$ with constant coefficients approximating $\mathbf {L}$ and $\mathbf {M}$. Again apriori bounds are derived for assumed solutions of the oblique boundary derivative problem for the purpose os selecting subsequences of sequences of approximating solutiions in the next chapter.