Conformal Parametrisation of Star-Shaped Domains and Persistence of Semilinear Elliptic Solutions
We study the conformal parametrisation of smooth star-shaped planar domains and its dependence on perturbations of the boundary. The boundary correspondence of the normalized Riemann map is described by the Theodorsen equation, which we formulate in weighted Wiener algebras. We derive an explicit representation of the inverse of the linearized Theodorsen operator through a canonical Wiener--Hopf factorisation and obtain quantitative a posteriori estimates that are stable under perturbations of the boundary function. For a collection of nonperturbative reference domains, rigorous computer-assisted estimates yield explicit neighborhoods in a weighted Wiener algebra such that every boundary function in these neighborhoods admits a uniquely determined conformal parametrisation in the validated ball, with quantitative control of the corresponding Riemann map. Thus the validation applies to open, infinite-dimensional families of star-shaped domains. As an application, we consider the semilinear Dirichlet problem \[ -Δv=v^3\quad\text{in }Ω, \qquad v=0\quad\text{on }\partialΩ. \] Pulling the equation back by the validated conformal maps yields a problem on the unit disk. Uniform a posteriori estimates then imply persistence of the elliptic solutions throughout explicit neighborhoods of the reference domains.