As is well known, the mere fact that a function of a complex variable is differentiable on a domain in the complex plane implies that the function has many other properties; for example, the real and imaginary parts satisfy the Cauchy-Riemann equations, the function is infinitely differentiable on the domain, etc. In addition, if a function on a domain Ω in Rn has continuous second partials and satisfies Laplace's equation thereon, then other properties follow as a consequence, such as the averaging property, etc. It will be shown in this chapter that there are limitations on the size of the H(b) 2+a(Ω) norm of a solution of the oblique boundary derivative problem for elliptic equations. These inequalities are known as apriori inequalities since it is assumed that a function is a solution without actually knowing that there are solutions. The establishment of such inequalities will pave the way for proving the existence of solutions in the next chapter.
Eventually, very strong conditions will have to be imposed on Ω. So strong, in fact, that the reader might reasonably conclude that only a spherical chip, or some topological equivalent, will satisfy all the conditions. As the ultimate application will involve spherical chips, the reader might benefit from assuming that Ω is a spherical chip. But as some of the inequalities require less stringent conditions on Ω, the conditions on Ω will be spelled out at the beginning of each section. Since these conditions will not be repeated in the formal statements, it is important to refer back to the beginning of each section for a description of Ω.
Unable to display preview. Download preview PDF.