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
We consider the inverse problem of localizing dipolar sources in an ellipsoid from boundary data, which we approach and constructively solve with techniques from harmonic and complex analysis. We use ellipsoidal harmonics to isolate the singular part of the solution, which we consider on a family of two-dimensional sections of the domain. We then use approximation theory to locate its singularities, and provide an algorithm which allows us to recover the sources from these singularities. We provide numerical illustrations related to the localization of pointwise dipolar sources in the human brain from electroencephalography (EEG).
Abstract. The Weinstein equation with complex coefficients is the equation governing generalized axisymmetric potentials (GASP) which can be written as Lm[u] = ∆u + (m/x) ∂xu = 0, where m ∈ C. We generalize results known for m ∈ R to m ∈ C. We give explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities, then we prove a Green's formula for GASP in the right half-plane H + for Re m < 1. We establish a new decomposition theorem for the GASP in any annular domains for m ∈ C, which is in fact a generalization of the Bôcher's decomposition theorem. In particular, using bipolar coordinates, we prove for annuli that a family of solutions for GASP equation in terms of associated Legendre functions of first and second kind is complete. For m ∈ C, we show that this family is even a Riesz basis in some non-concentric circular annulus.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.