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The $\overline\partial$-Equation on An Annulus with Mixed Boundary Conditions
by Xiaoshan Li   Mei-Chi Shaw

Vol. 8 No. 3 (2013) P.399~P.411

ABSTRACT

In this paper we study the $\overline\partial$-equation with mixed boundary conditions on an annulus $\Omega=\Omega_1\setminus\overline \Omega_2\subset\subset \mathbb{C}^n$ between two pseudoconvex domains satisfying $\Omega_2\subset\subset\Omega_1$. We prove $L^2$-existence theorems for $\overline\partial_{\rm mix}$ for any $\overline\partial_{\rm mix}$-closed $(p,q)$-form with $2\leq q\leq n$. For the critical case when $q=1$ on the annulus $\Omega$, we shwo that the space of harmonic forms is infinite dimensional and $H^{(p,1)}_{\overline\partial_{\rm mix}, L^2}(\Omega)$ is isomorphic to the quotient of $H_{W^1}^{(p,0)}(\Omega_2)$ with coefficients in $W^1(\Omega_2)$ over the Bergman space $H_{L^2}^{(p,0)}(\Omega_1)$ on the pseudoconvex domain $\Omega_1$. Boundary regularity for the corresponding operators is also obtained.

KEYWORDS
Cauchy-Riemann equations, pseudo-concave domains, $\overline\partial$-Dirichlet condition, $\overline\partial$-Neumann condition

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 32W05, 35N15, 58J32

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