R{\'e}solution du $\partial \bar{\partial}$ pour les formes diff{\'e}rentielles ayant une valeur au bord au sens des courants d{\'e}finies sur un domaine L{\'e}vi-plat non born{\'e} de} $\mathbb{C}^{n}$

TL;DR

This paper addresses the complex analysis problem of solving the $ar{ ext{d}}$-equation for differential forms with boundary values in the sense of currents on unbounded Levi-flat domains in complex Euclidean space.

Contribution

It provides a solution to the $ar{ ext{d}}$-problem for forms with distribution boundary values on unbounded Levi-flat domains, extending previous results to non-compact settings.

Findings

Solved the $ar{ ext{d}}$-problem for forms with boundary values in the sense of currents

Extended the theory to unbounded Levi-flat domains in $ ext{C}^n$

Demonstrated the complementary domain is also Levi-flat and unbounded

Abstract

We solve the $\partial \bar{\partial}$-problem for a form with distribution boundary value on a Levi flat unbounded domain of $\mathbb{C}^n$ with the complementary is also Levi flat and unbounded.