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Solutions of Pascali systems attached to convex boundaries

Barbara Drinovec Drnovšek, Uroš Kuzman

TL;DR

The paper proves that for a bounded strictly convex domain $\Omega\Subset \mathbb{C}^n$ and a point $q\in\Omega$, there exists a continuous Pascali-system solution $u\in \mathcal{O}_B(\overline{\mathbb{D}},\mathbb{C}^n)$ centered at $q$ with $u(\mathbb{D})\subset \Omega$ and $u(\partial\mathbb{D})\subset \partial\Omega$. It develops a robust functional-analytic framework for Pascali systems, including a bounded right inverse $Q_B$ and an invertible $\widehat{\Psi}$ linking solvability to holomorphy, and then combines this with a nonlinear Riemann–Hilbert construction to deform boundary data. The main result is obtained via a two-stage process: (i) create a small centered solution inside $\Omega$, and (ii) iteratively push the boundary toward $\partial\Omega$ along tangent directions to convex level sets of a defining function $\rho$, ensuring convergence to a continuous, boundary-to-boundary map. The work extends complex-analytic, boundary-value techniques to Pascali systems and yields a geometric, proper-map analogue for these elliptic systems, with potential implications for boundary-value problems in several complex variables and related geometric analysis.

Abstract

Given a bounded strictly convex domain $Ω\Subset \mathbb{C}$ and a point $q\in Ω$ we construct a continuous solution of the Pascali-type elliptic system of differential equations that is centered in $q$, maps the unit disc into $Ω$ and the unit circle into $\partial Ω$.

Solutions of Pascali systems attached to convex boundaries

TL;DR

The paper proves that for a bounded strictly convex domain and a point , there exists a continuous Pascali-system solution centered at with and . It develops a robust functional-analytic framework for Pascali systems, including a bounded right inverse and an invertible linking solvability to holomorphy, and then combines this with a nonlinear Riemann–Hilbert construction to deform boundary data. The main result is obtained via a two-stage process: (i) create a small centered solution inside , and (ii) iteratively push the boundary toward along tangent directions to convex level sets of a defining function , ensuring convergence to a continuous, boundary-to-boundary map. The work extends complex-analytic, boundary-value techniques to Pascali systems and yields a geometric, proper-map analogue for these elliptic systems, with potential implications for boundary-value problems in several complex variables and related geometric analysis.

Abstract

Given a bounded strictly convex domain and a point we construct a continuous solution of the Pascali-type elliptic system of differential equations that is centered in , maps the unit disc into and the unit circle into .
Paper Structure (2 sections, 6 theorems, 31 equations)

This paper contains 2 sections, 6 theorems, 31 equations.

Table of Contents

  1. On solutions of Pascali systems
  2. Proof of Theorem \ref{['main']}

Key Result

Theorem 1

Let $\Omega\Subset \mathbb{C}^n$ be a smoothly bounded strictly convex domain and $q\in \Omega$. There exists a map $u\in \mathcal{O}_B(\overline{\mathbb{D}},\mathbb{C}^n)$ centered in $q$ and such that $u(\mathbb{D})\subset \Omega$, $u(\partial\mathbb{D})\subset \partial\Omega$. In particular, the

Theorems & Definitions (10)

  • Theorem 1
  • Proposition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Lemma 6
  • proof