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William L. Blair

Publications and source records attributed to William L. Blair.

8 recordsLinked to original sources
arXiv2026

On the Existence of Solutions to the Seiberg-Witten Vortex Equations with Exponential Decay on the Plane

Clifford Taubes showed that the moduli space of the variational equations of the Yang-Mills-Higgs functional on the plane is non-empty, and its elements correspond to "vortices". Inspired by this result, in this paper, we completely characterize exponential decay solutions to the Seiberg-Witten vortex equations, where the equations are derived by applying a Hitchin-type dimensional reduction. We show that the existence of Seiberg-Witten ``vortices'' is not provided by the exponential decay solutions. This is in contrast to the Yang-Mill-Higgs vortex equations.

math.AP↗
arXiv2026

Generalizations of the Dirichlet problem for bianalytic functions

We prove the Dirichlet problem for second-order iterated Vekua equations, a natural generalization of the Bitsadze equation, is well-posed when the boundary condition is defined as a product of an exponential function and a polynomial on a non-degenerate conic that is not a circumference. Also, we extend this result, as well as other related results for the Dirichlet problem for polyanalytic and generalized analytic functions from the literature, to their analogues for bicomplex differential equations.

math.AP↗
arXiv2025

Bicomplex Hardy Classes of Solutions to Beltrami Equations and the Schwarz Boundary Value Problem

We define Hardy classes of bicomplex-valued functions on the complex unit disk which solve bicomplex versions of the Beltrami and related equations. Using representations in terms of their complex-valued counterparts, we show these bicomplex-valued functions recover the boundary behavior associated with the classic holomorphic Hardy spaces. This work generalizes known results for complex-valued functions and continues recent work in the setting of bicomplex analogues of Hardy spaces of both holomorphic and generalized analytic functions. Also, we show Schwarz and Dirichlet boundary value problems associated with the bicomplex Beltrami equation are solvable and provide solution formulas.

math.CV↗
arXiv2025

An Atomic Representation for Bicomplex Hardy Classes

We develop representations for bicomplex-valued functions in Hardy classes that generalize the complex holomorphic Hardy spaces. Using these representations, we show these functions have boundary values in the sense of distributions that are representable by an atomic decomposition, and we show continuity of the Hilbert transform on this class of distributional boundary values.

math.CV↗
arXiv2023

Hardy Spaces of Meta-Analytic Functions and the Schwarz Boundary Value Problem

We extend representation formulas that generalize the similarity principle of solutions to the Vekua equation to certain classes of meta-analytic functions. Also, we solve a generalization of the higher-order Schwarz boundary value problem in the context of meta-analytic functions with boundary conditions that are boundary values in the sense of distributions.

math.CV↗
arXiv2023

An atomic representation for Hardy classes of solutions to nonhomogeneous Cauchy-Riemann equations

We develop a representation of the second kind for certain Hardy classes of solutions to nonhomogeneous Cauchy-Riemann equations and use it to show that boundary values in the sense of distributions of these functions can be represented as the sum of an atomic decomposition and an error term. We use the representation to show continuity of the Hilbert transform on this class of distributions and use it to show that solutions to a Schwarz-type boundary value problem can be constructed in the associated Hardy classes.

math.CV↗