Proof of Theorem 1.1

Abstract

This chapter is devoted to the proof of Theorem 1.1. The idea of our proof is stated as follows. First, we reduce the study of the boundary value problem

$$ \left\{ \begin{array}{l} ({\rm A - }\lambda {\rm )u = f }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm in D, } \\ {\rm Lu = }\mu {\rm (x')}\frac{{\partial {\rm u}}}{{\partial {\rm n}}} + \Upsilon (x')u = \varphi \,on\,\partial D \\ \end{array} \right. $$

(1.1)

to that of a first-order pseudo-differential operator T(λ) = LP(λ) on the boundary ∂D, just as in Section 4.3. Then we prove that conditions (A) and (B) are sufficient for the validity of the a priori estimate

$$\parallel u\parallel _{2,p} \le C(\lambda )(\parallel f\parallel _p + |\varphi| _{2 - 1/p,p} + \parallel u\parallel _p ).$$

(1.2)