Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in $${\mathbb {C}}{\mathbb {P}}^n$$ C P n

Harrington, Phillip S.

doi:10.1007/s12220-017-9809-0

Cited by 9 publications

(5 citation statements)

References 33 publications

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“…Plantations currently cover about 10% of the region. The topography, climate, land use, geology, hydrogeology, and soils of the region are described in detail by Harrington et al (2011), Harrington et al (2013), SENRMB (2019) and summarized in Crosbie et al (2015). In summary, the study area is a gently undulating coastal plain of marine origin located in the southeast corner of South Australia, bounded by the Southern Ocean on the west and south sides, the South Australia/Victoria border on the east side and the Murray‐Darling Basin to the north.…”

Section: Methodsmentioning

confidence: 99%

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Effects of climate variability and change on groundwater impacts of forestry plantations

Benyon,

Doody,

Lawson

et al. 2024

Hydrological Processes

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Quantifying water use of various water consumers is an essential part of sustainable water management. Annual evapotranspiration (ET) of plantation forests often exceeds that of dryland agriculture, which in South Africa and South Australia has resulted in restrictions on plantation development. In the latter case, water licences are issued to commercial forestry plantations to account for higher ET compared to dryland pasture. Unlike irrigated crops, it is not practicable to measure water use of plantations directly and so in South Australia a set of ‘deemed’ average water use rates has been applied since 2013, based on species and depth to groundwater. Since South Australia's ‘deemed’ rates were calculated, additional plot‐scale measurements of annual ET from plantations <2 years old and post‐canopy closure have been used to quantify various components of ET. This has enabled development of two empirical ET models for plantations in South Australia's Lower Limestone Coast, and facilitated an advanced understanding of the effect of plantations on hydrological processes, particularly in relation to groundwater use. In this study, we applied these models to estimate rotation‐averaged annual ET and net groundwater impacts (net groundwater extraction plus recharge reduction compared to pasture) of plantations, driven by climate and groundwater depth, for comparison with the deemed rates. The modelling suggests that the groundwater impacts of plantations vary in space and time and that the deemed rates over‐estimate these impacts, on average. Accounting for variation in the effects of climate on the various components of ET, both spatially and temporally, may allow for more flexible rules for water resource allocation than using any simple, rule‐of‐thumb approach.

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Section: Methodsmentioning

confidence: 99%

“…The topography, climate, land use, geology, hydrogeology, and soils of the region are described in detail by Harrington et al (2011), Harrington et al (2013, SENRMB (2019) and summarized in Crosbie et al (2015).…”

Section: Description Of the Study Region And Current Water Licensing ...mentioning

confidence: 99%

Effects of climate variability and change on groundwater impacts of forestry plantations

Benyon,

Doody,

Lawson

et al. 2024

Hydrological Processes

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“…In this paper, the φ is Hölder continuous on the boundary ∂Ω. In [13], he also obtained some results about pseudoconvex domain with Lipschitz boundary in CP n .…”

Section: Introductionmentioning

confidence: 94%

The Intrinsic Geometry on Bounded Pseudoconvex Domains

Liu

2017

J Geom Anal

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The Diederich-Fornaess index has been introduced since 1977 to classify bounded pseudoconvex domains. In this article, we derive several intrinsic, geometric conditions on boundary of domains for arbitrary indexes. Many results, in the past, by various mathematicians estimated the index by assuming some properties of domains. Our motivation of this paper is, the other way around, to look for how the index effects properties and shapes of domains. Especially, we look for a necessary condition of all bounded pseudoconvex domains Ω ⊂ C 2 with the DiederichFornaess index 1. We also show that, when the Levi-flat set of ∂Ω is a closed Riemann surface, then the necessary condition can be simplified. IntroductionLet Ω be a bounded domain in C n with smooth boundary andThe Ω is said to be pseudoconvex if − log(−δ(z)) is plurisubharmonic in Ω. Note that − log(−δ(z)) blows up whenever z approaches the boundary ∂Ω. Indeed, all bounded pseudoconvex domains with C 2 boundary admit bounded (strictly) plurisubharmonic functions which vanishes on ∂Ω was shown by . They proved that any relatively compact pseudoconvex domain in Stein manifolds admits a defining function ρ such that −(−ρ) η is (strictly) plurisubharmonic in Ω for some η ∈ (0, 1]. The author also remark that the ρ may not be δ in general. For the pseudoconvex domain in complex manifolds, see Range [21].For the bounded pseudoconvex domain Ω with C 1 boundary, Kerzman-Rosay constructed a smooth (strictly) plurisubharmonic function in [17]. This function also vanishes on the boundary ∂Ω. Later, Demailly in [8] improved the result of Kerzman-Rosay to arbitrary bounded pseudoconvex domains in C n with Lipschitz boundary. He also showed his plurisubharmonic smooth function φ is bounded above and below by a multiple of − 1 log(−δ) near the boundary. In [14], Harrington found a new smooth (strictly) plurisubharmonic φ in bounded pseudoconvex domains in C n with Lipschitz boundary. In this paper, the φ is Hölder continuous on the boundary ∂Ω. In [13], he also obtained some results about pseudoconvex domain with Lipschitz boundary in CP n .If the boundary is smooth or at least C 2 , then the above result of Diederich-Fornaess is applicable. In particular, the fact that −(−ρ) η 0 is (strictly) plurisubharmonic will implies that −(−ρ) η is

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“…This follows from the existence of bounded plurisubharmonic functions on pseudoconvex domains in CP n with C 2 boundary (see [13]). This is even true if D has only Lipschitz boundary (see [5]).…”

Section: Proof Let Us Consider the Function H Define On Hmentioning

confidence: 99%

Solving $\overline\partial$ with prescribed support on Hartogs triangles in $\mathbb C^2$ and $\mathbb C\mathbb P^2$

Laurent-Thiébaut¹,

Shaw²

2016

Preprint

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In this paper we consider the problem of solving the Cauchy-Riemann equation with prescribed support. More precisely, let X be a complex manifold of complex dimension n and Ω ⊂ X a subdomain of X. We ask the following questions:Let T be a ∂-closed (r, 1)-current, 0 ≤ r ≤ n, on X with support contained in Ω, does there exist a (r, 0)-current on X, with support contained in Ω, such that ∂S = T ?If moreover T = f is a smooth form or a C k form or an L p loc form, can we find g with support contained in Ω and with the same regularity as f such that ∂g = f ?This leads us to introduce the Dolbeault cohomology groups with prescribed support in Ω. Let us denote by H r,1 Ω,∞ (X) the quotient spaceIn the same way, we define H r,1 Ω,C k (X), H r,1 Ω,L p loc (X) and H r,1 Ω,cur (X) for the C k , L p loc and the current category.The cohomology groups H r,1 Ω,∞ (X), H r,1 Ω,C k (X) ,H r,

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Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in $${\mathbb {C}}{\mathbb {P}}^n$$ C P n

Cited by 9 publications

References 33 publications

Effects of climate variability and change on groundwater impacts of forestry plantations

Effects of climate variability and change on groundwater impacts of forestry plantations

The Intrinsic Geometry on Bounded Pseudoconvex Domains

Solving $\overline\partial$ with prescribed support on Hartogs triangles in $\mathbb C^2$ and $\mathbb C\mathbb P^2$

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