Neumann Domains on Quantum Graphs

The Neumann Problem on Lipschitz Domains

Au — 0 in D; u = ƒ on bD9 where ƒ and its gradient on 3D belong to L(do). For C domains, these estimates were obtained by A. P. Calderón et al. [1]. For dimension 2, see (d) below. In [4] and [5] we found an elementary integral formula (7) and used it to prove a theorem of Dahlberg (Theorem 1) on Lipschitz domains. Unknown to us, this formula had already been discovered long ago by Payne and We...

On Neumann Eigenfunctions in Lip Domains

A planar set D will be called a lip domain if it is Lipschitz, open, bounded, connected, and given by (1) D = {(x1, x2) : f1(x1) < x2 < f2(x1)}, where f1, f2 are Lipschitz functions with constant 1. The assumption that D is a Lipschitz domain puts an extra constraint on the functions fk; we discuss this issue in greater detail later in this section. Let μ2 denote the second eigenvalue for the L...

∂̄b-NEUMANN PROBLEM ON NONCHARACTERISTIC DOMAINS

We study the ∂̄b-Neumann problem for domains Ω contained in a strictly pseudoconvex manifold M2n+1 whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts of a single CR function w. When the Kohn Laplacian is a priori known to have closed range in L2, we prove sharp regularity and estimates for solutions. We establish a condition on the...

Quantum Hall Dynamics on Von Neumann Lattice

h in quantum Hall regime of the system of interactions and disorders. This shows that a determination of the fine structure constant from integer quantum Hall effect is in fact possible. We present also a unified mean field theory of the fractional Hall effect for the in-compressible quantum liquid states based on flux condensation and point out that the known Hofstadter butterfly spectrum of t...

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras