Pseudo-Differential Operator Approach to Agmon’s Method

In this chapter, by using Shmuel Agmon’s method (Agmon, Lectures on Elliptic Boundary Value Problems, Revised edition of the 1965 original. AMS Chelsea Publishing, Providence, 2010) we study the homogeneous Robin problem with a complex parameter \(\lambda \) under the conditions (H.1) and (H.2). In the framework of \(L^{p}\) Sobolev spaces, we associate with this spectral problem a densely defined, closed linear operator \(\mathfrak {A}_{p}\) in the Banach space \(L^{p}(\varOmega )\) . In Sect. 10.1 we prove fundamental global a priori estimates for \(\mathfrak {A}_{p} - \lambda I\) with a complex parameter \(\lambda \) , which plays an important role in the proof of our generation theorem for analytic semigroups. In the proof we make use of Agmon’s idea in the proof of the surjectivity of \(\mathfrak {A}_{p} - \lambda I\) . More precisely, in order to prove an existence and uniqueness theorem for the homogeneous Robin boundary value problem in the framework of \(L^{p}\) Sobolev spaces when \(\vert \lambda \vert \) tends to \(\infty \) , we make use of a method essentially due to Agmon. This is a technique of treating a spectral parameter \(\lambda \) as a second order, elliptic differential operator of an extra variable y on the unit circle S, and relating the old problem to a new one with the additional variable. First, following Fujiwara (J. Fac. Sci. Univ. Tokyo Sec. IA 17:123–152, 1970) we replace the complex parameter \(\lambda \) by a second order differential operator on S. We express the complex parameter \(\lambda \) in the form \(\lambda = r^{2}\,e^{i\theta }\) for \(r \geq 0\) and \(-\pi < \theta < \pi \) , and replace the original differential operator \(A - \lambda = A - r^{2}\,e^{i\theta }\) defined in the original domain \(\varOmega \) by the second order differential operator \(\widetilde {\Lambda }(\theta ) = A + e^{i\theta }\,{\partial ^{2}}/{\partial y^{2}}\) defined in the product domain \(\varOmega \times S\) with the boundary \({\partial \varOmega } \times S\) . Secondly, we consider instead of the original Robin boundary value problem with spectral parameter the homogeneous Robin boundary value problem in the product domain \(\varOmega \times S\) . Thirdly, we associate with this augmennted homogeneous Robin boundary value problem the augmented closed realization \(\widetilde {\mathfrak {A}}_{p}(\theta )\) in the Banach space \(L^{p}(\varOmega \times S)\) . Then we can prove that the operator \(\widetilde {\mathfrak {A}}_{p}(\theta )\) is a Fredholm operator (see Theorem 10.1).