Generation Theorem for Analytic Semigroups via Agmon’s Method

Let \(\varOmega \) be a relatively compact, n-dimensional smooth manifold with boundary \(\partial \varOmega \) . In this chapter we prove Theorem 1.1 (Theorem 11.2) under the conditions (H.1) and (H.2). More precisely, we study the homogeneous Robin problem with a complex parameter \(\lambda \) , and associate with the spectral Robin problem a densely defined, closed linear operator \(\mathfrak {A}_{p}\) in the Banach space \(L^{p}(\varOmega )\) . In the proof of Theorem 11.2 we make use of Agmon’s method (Theorem 10.1 and Proposition 11.7) developed in Chap. 10 . In this way, we can prove that the desired a priori estimate ( 1.5 ) in Theorem 1.1 holds true.