Dirichlet and Neumann eigenvalue problems on CR manifolds

Abstract

We study the properties of Carnot–Caratheodory spaces attached to a strictly pseudoconvex CR manifold M, in a neighborhood of each point $$x \in M$$ , versus the pseudohermitian geometry of M arising from a fixed positively oriented contact form $$\theta $$ on M. The weak Dirichlet problem for the sublaplacian $$\Delta _b$$ on $$(M, \theta )$$ is solved on domains $$\Omega \subset M$$ supporting the Poincare inequality. The solution to Neumann problem for the sublaplacian $$\Delta _b$$ on a $$C^{1,1}$$ connected $$(\epsilon , \delta )$$ -domain $$\Omega \subset {{\mathbb {G}}}$$ in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of $$\Delta _b$$ on $$\Omega \subset M$$ in a Carnot–Cartheodory complete pseudohermitian manifold $$(M, \theta )$$ .