Global regularity of the ∂-Neumann problem

Catlin, David W.

doi:10.1090/pspum/041/740870

Cited by 101 publications

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“…Let f δ be defined by (1). By Proposition 5, f δ ∈ C ∞ (Ω) converge to f in the Λ α (Ω) topology as δ goes to zero.…”

Section: Approximation By Holomorphic Functionsmentioning

confidence: 97%

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On the Mergelyan approximation property on pseudoconvex domains in ℂⁿ

Cho

1998

Proc. Amer. Math. Soc.

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“…Let f δ be defined by (1). By Proposition 5, f δ ∈ C ∞ (Ω) converge to f in the Λ α (Ω) topology as δ goes to zero.…”

Section: Approximation By Holomorphic Functionsmentioning

confidence: 97%

“…For each fixed δ > 0, let us solve ∂u δ = ∂f δ on Ω. Then by Catlin's global regularity theorem for the ∂-equation on pseudoconvex domains of finite type in C n [1], it follows that u δ ∈ C ∞ (Ω). Also, by (3), u δ satisfies…”

Section: Approximation By Holomorphic Functionsmentioning

confidence: 99%

On the Mergelyan approximation property on pseudoconvex domains in ℂⁿ

Cho

1998

Proc. Amer. Math. Soc.

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“…The following assertion is essentially a localized version of Catlin's result [7] that property (P) implies global regularity.…”

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confidence: 90%

“…Catlin [7] also found a more general condition, property (P), guaranteeing global regularity of the (9-Neumann problem. This property says that there exist bounded plurisubharmonic functions having all eigenvalues of their complex Hessian arbitrarily large at the boundary.…”

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confidence: 97%

Small Sets of Infinite Type are Benign for the $\overline\partial$-Neumann Problem

Boas¹

1988

Proceedings of the American Mathematical Society

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ABSTRACT. An explicit construction shows that the 3-Neumann operator and the Bergman and Szegô projections are globally regular in every smooth bounded pseudoconvex domain whose set of boundary points of infinite type has Hausdorff two-dimensional measure equal to zero. On the other hand there are examples of domains with globally regular d-Neumann operator but whose infinite-type points fill out an open subset of the boundary.Is the ¿9-Neumann problem globally regular on every smooth bounded pseudoconvex domain in C"? Although this basic question remains open, some sufficient geometric conditions on the domain for an affirmative answer are known. For instance the answer is yes if the boundary of the domain repels analytic discs in a suitable differential-geometric sense.Here I show that the answer is yes if the boundary of the domain excludes analytic discs in a suitable measure-theoretic sense. This result was recently obtained by Sibony [25] as a corollary of a general theory. What is new in this paper is the simple proof, an elementary and explicit construction.It was a quarter of a century ago that Kohn [17] made the first breakthrough on the 9-Neumann problem by establishing both global and local regularity in case the domain is strongly pseudoconvex. More recently Catlin [8] proved that a subelliptic estimate holds near a boundary point p if and only if p is a point of finite type in the sense of D'Angelo [11], meaning that complex varieties have bounded order of contact with the boundary at p. Consequently the ¿5-Neumann problem is locally regular near points of finite type, and so if obstructions to global regularity exist then they must lie in the set of boundary points of infinite type.Catlin [7] also found a more general condition, property (P), guaranteeing global regularity of the (9-Neumann problem. This property says that there exist bounded plurisubharmonic functions having all eigenvalues of their complex Hessian arbitrarily large at the boundary. Catlin proved that property (P) is satisfied by domains that are regular in the sense of Diederich and Fornsess [13, 15]. For instance a domain whose weakly pseudoconvex boundary points lie in a totally real submanifold is regular, but a domain that contains a complex disc in its boundary is not.Subsequently Sibony [25] introduced the notion of S-regularity. The boundary of a smooth bounded pseudoconvex domain is ß-regular if every continuous function on the boundary is the boundary value of a function plurisubharmonic

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“…(2) For each w g ft, the linear space Ran(T^_w) defined by Ran(^_",) = j J2(<pj -Wj)fj : f £ H2(U), 1 < j < n I is closed in H2 and has codimension m. domains that are not of finite type in general, but that are still pseudoregular, are domains satisfying Catlin's property P (see [4]). Typically, pseudoconvex Reinhardt domains with no analytic disks in their boundaries are pseudoregular (see [21,19]).…”

Section: 5])mentioning

confidence: 99%