On regularity of ∂-solutions on a_q domains with C² boundary in complex manifolds

2022/05/24 by Xianghong Gong, Gong, Xianghong
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2205.12025

Abstract

We study regularity of solutions u to ∂ u=f on a relatively compact C² domain D in a complex manifold of dimension n, where f is a (0,q) form. Assume that there are either (q+1) negative or (n-q) positive Levi eigenvalues at each point of boundary ∂ D. Under the necessary condition that a locally L² solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain 1/2 derivative when q=1 and f is in the Hölder-Zygmund space Λ^r( D) with r>1. For q>1, the same regularity for the solutions is achieved when ∂ D is either sufficiently smooth or of (n-q) positive Levi eigenvalues everywhere on ∂ D.

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