Hypoelliptic Robin Problems via Boutet de Monvel Calculus

This chapter is devoted to a functional analytic approach to the study of the hypoelliptic Robin problem for a second order, strictly elliptic differential operator with a complex parameter \(\lambda \) . We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous Robin problem when \(\vert \lambda \vert \) tends to \(\infty \) (Theorem 12.3). Our proof is based on Agmon’s theorem (Theorem 7.6 ) and Remark 7.7 . More precisely, we prove the spectral properties of the closed realization \(\mathfrak {A}_{2}\) of the strictly elliptic differential operator, similar to the elliptic (non-degenerate) case. However, in the degenerate case we cannot use Green’s formula to characterize the adjoint operator \({\mathfrak {A}_{2}}^{\ast }\) of the closed realization in the framework of \(L^{2}\) Sobolev spaces. Hence, we shift our attention to its resolvent. In Sect. 12.8 we characterize the resolvent \(\mathcal {G}\) of the homogeneous Robin problem in terms of the Boutet de Monvel calculus (Theorem 12.20 and Corollary 12.21). Although the Fredholm boundary operator T is not elliptic or degenerate on the boundary, the original Boutet de Monvel calculus (Acta Math. 126:11–51, 1971) goes through smoothly. Indeed, it suffices to note that the pseudo-differential operator T has a parametrix S in the Hörmander class under the conditions (H.1) and (H.2). In Sect. 12.9 we characterize the mapping property of the adjoint \(\mathcal {G}^{\ast }\) of the resolvent in the framework of Sobolev spaces of \(L^{p}\) type (Theorem 12.22). This chapter has its own right from the point of view of spectral analysis.