2003
DOI: 10.1512/iumj.2003.52.2261
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Solution of the Neumann problem on a non-smooth domain

Abstract: Abstract. We study the solution of the∂-Neumann problem on (0, 1)-forms on the product of two half-planes in C 2 . In, particular, we show the solution can be decomposed into functions smooth up to the boundary and functions which are singular at the singular points of the boundary. Furthermore, we show the singular functions are log and arctan terms.

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Cited by 7 publications
(21 citation statements)
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“…We continue here the study of the problem for (0, 1)-forms on model domains, focusing on the bi-disc, Ω = D 1 × D 2 ∈ C 2 , where D 1 ⊂ C and D 2 ⊂ C are defined by the equations r 1 < 1 and r 2 < 1, respectively, where r j = |z j |, j = 1, 2. The existence of a solution in L 2 (Ω) is given by Hörmander [6]. We shall see singularities only occur on the distinguished boundary, ∂D 1 × ∂D 2 .…”
Section: Introductionmentioning
confidence: 85%
See 1 more Smart Citation
“…We continue here the study of the problem for (0, 1)-forms on model domains, focusing on the bi-disc, Ω = D 1 × D 2 ∈ C 2 , where D 1 ⊂ C and D 2 ⊂ C are defined by the equations r 1 < 1 and r 2 < 1, respectively, where r j = |z j |, j = 1, 2. The existence of a solution in L 2 (Ω) is given by Hörmander [6]. We shall see singularities only occur on the distinguished boundary, ∂D 1 × ∂D 2 .…”
Section: Introductionmentioning
confidence: 85%
“…then u 1 and v 1 are related by The integral in (4.7) was considered in [2] and gives, after summing over m 1 and m 2 , and using a theorem of Borel, with similar results on the form of u 2 , Theorem 1.1.…”
Section: Behavior At the Distinguished Boundarymentioning
confidence: 97%
“…Other results in this direction belong to Ehsani [9], [10]. The key ingredient in the proof of all of the results above is an exhaustion of the piecewise smooth domain by smooth (or uniformly Lipschitz) strictly pseudoconvex domains Ω on which the ∂-Neumann operator exists and satisfies uniform L 2 or subelliptic ε-estimates.…”
Section: Introduction and Resultsmentioning
confidence: 96%
“…The study of the ∂ and ∂-Neumann problems on product domains raises a series of interesting questions, which have been studied by many authors [8,9,10,1,3,12]. In [12], the method of separation of variables was used to compute the spectrum of the complex Laplacian = ∂∂ * + ∂ * ∂ on a polydisc, and for each eigenvalue, the corresponding eigenspace was identified.…”
Section: Introductionmentioning
confidence: 99%