2013 / September Volume 8 No.3
The $\overline\partial$-Equation on An Annulus with Mixed Boundary Conditions
| Published Date |
2013 / September
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|---|---|
| Title | The $\overline\partial$-Equation on An Annulus with Mixed Boundary Conditions |
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| Pagination | 399-411 |
| 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.
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| AMS Subject Classification |
32W05, 35N15, 58J32
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| Received |
2013-09-01
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| Accepted |
2013-09-01
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