Hardy spaces of the conjugate Beltrami equation

Abstract

We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents $1<\infty$. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation ${div}(\sigma\nabla u)=0$ with $L^p$-boundary data

Hardy spaces for the conjugated Beltrami equation in a doubly connected domain

“…with ess sup The fact that f solves (1) and satisfies (3) entails that f has a non-tangential limit almost everywhere on T 1 , denoted by tr f , and the trace in (2) has to be understood in this sense. Moreover, f is unique up to a purely imaginary constant, and if we normalize f by 2π 0 Im tr f e iθ dθ = 0, then f is unique and…”

“…An important difference with the case of simply connected domains, due to the fact that the boundary has now two connected components, is that, in the Dirichlet problem, we prescribe the real part of the solution on one part of the boundary and the imaginary part on the other. Another difference with [2] is that we only assume that ν ∈ W 1,q R (G 2 ) for some q ∈ (2, +∞] instead of being Lipschitz continuous.…”

“…There are two main differences with previous studies (Baratchart et al, 2010 [2]). First, while the simple connectivity plays an important role in Baratchart et al (2010) [2], the multiple connectivity of the domain leads to unexpected difficulties. In particular, we make strong use of a suitable parametrization of an analytic function in a ring by its real part on one part of the boundary and by its imaginary part on the other.…”

“…In particular, we make strong use of a suitable parametrization of an analytic function in a ring by its real part on one part of the boundary and by its imaginary part on the other. Then, we allow the coefficient in the conjugated Beltrami equation to belong to W 1,q for some q ∈ (2, +∞], while it was supposed to be Lipschitz in Baratchart et al (2010) [2]. We define Hardy spaces associated with the conjugated Beltrami equation and solve the corresponding Dirichlet problem.…”

Strain-induced topological insulator phase and effective magnetic interactions in Li2IrO3

“…3,38,39 Our previous work on Na 2 IrO 3 using the local-density approximation (LDA) + U method has reported a Neel-type collinear antiferromagnetic-ordered ground state with fully gapped bands, with its magnetic moments parallel to the Ir honeycomb plane without any favoring direction inside the layer. 40 In contrast, several experimental observations and first-principle calculations 27,28,41 independently suggest a zigzag-type antiferromagnetic order (zigzag-AF) in-plane ordering of spin- 1 2 moments for the same material. Theoretical investigations had been concentrated on explaining this magnetic ground state within the framework of the HK model.…”