Global solvability and regularity for ∂¯ on an annulus between two weakly convex domains which satisfy property (P)

2019 ◽  
Vol 12 (03) ◽  
pp. 1950041
Author(s):  
Sayed Saber
Keyword(s):  
Neumann Problem ◽  
Complex Manifold ◽  
Global Solvability ◽  
Closed Formula ◽  
Convex Domains ◽  
Weakly Convex ◽  
Canonical Solution ◽  
Dimension Formula ◽  
Satisfy Property

Let [Formula: see text] be a complex manifold of dimension [Formula: see text] and let [Formula: see text]. Let [Formula: see text] be a weakly [Formula: see text]-convex and [Formula: see text] be a weakly [Formula: see text]-convex in [Formula: see text] with smooth boundaries such that [Formula: see text]. Assume that [Formula: see text] and [Formula: see text] satisfy property [Formula: see text]. Then the compactness estimate for [Formula: see text]-forms [Formula: see text] holds for the [Formula: see text]-Neumann problem on the annulus domain [Formula: see text]. Furthermore, if [Formula: see text] is [Formula: see text]-closed [Formula: see text]-form, which is [Formula: see text] on [Formula: see text] and which is cohomologous to zero on [Formula: see text], the canonical solution [Formula: see text] of the equation [Formula: see text] is smooth on [Formula: see text].

Chern-Yamabe problem and Chern-Yamabe soliton

2021 ◽  
Vol 32 (03) ◽  
pp. 2150016
Author(s):  
Pak Tung Ho ◽  
Jinwoo Shin
Keyword(s):  
Scalar Curvature ◽  
Complex Manifold ◽  
The Other ◽  
Compact Complex Manifold ◽  
Yamabe Problem ◽  
Conformal Metric ◽  
Hermitian Metric ◽  
Dimension Formula ◽  
Complex Dimension ◽  
Yamabe Soliton

Let [Formula: see text] be a compact complex manifold of complex dimension [Formula: see text] endowed with a Hermitian metric [Formula: see text]. The Chern-Yamabe problem is to find a conformal metric of [Formula: see text] such that its Chern scalar curvature is constant. In this paper, we prove that the solution to the Chern-Yamabe problem is unique under some conditions. On the other hand, we obtain some results related to the Chern-Yamabe soliton.

scholarly journals Compactness of the ∂-Neumann Problem on Convex Domains

1998 ◽  
Vol 159 (2) ◽  
pp. 629-641 ◽  
Author(s):  
Siqi Fu ◽  
Emil J Straube
Keyword(s):  
Neumann Problem ◽  
Convex Domains ◽  
The Neumann Problem

scholarly journals APPROXIMATION OF HOLOMORPHIC MAPPINGS ON 1-CONVEX DOMAINS

2013 ◽  
Vol 24 (14) ◽  
pp. 1350108 ◽  
Author(s):  
KRIS STOPAR
Keyword(s):  
Complex Manifold ◽  
Convex Domain ◽  
Holomorphic Mappings ◽  
Holomorphic Maps ◽  
Convex Domains ◽  
Exceptional Set ◽  
Proper Holomorphic Maps ◽  
Pseudoconvex Boundary ◽  
Oka Principle ◽  
Additional Assumptions

Let π : Z → X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D ⋐ X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over [Formula: see text] which has prescribed core, it fixes the exceptional set E of D, and is dominating on [Formula: see text]. Each section in this spray is of class [Formula: see text] and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class [Formula: see text] and holomorphic on D can be approximated in the [Formula: see text] topology by π-sections that are holomorphic in open neighborhoods of [Formula: see text]. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.

scholarly journals The Neumann problem in Lp on Lipschitz and convex domains

2008 ◽  
Vol 255 (7) ◽  
pp. 1817-1830 ◽  
Author(s):  
Aekyoung Shin Kim ◽  
Zhongwei Shen
Keyword(s):  
Neumann Problem ◽  
Convex Domains ◽  
The Neumann Problem

Lp Neumann problem for some Schrödinger equations in (semi-)convex domains

2019 ◽  
Vol 22 (02) ◽  
pp. 1950007
Author(s):  
Sibei Yang ◽  
Dachun Yang
Keyword(s):  
Neumann Problem ◽  
Convex Domain ◽  
Boundary Data ◽  
Muckenhoupt Weight ◽  
Weight Class ◽  
Convex Domains ◽  
Reverse Hölder Class ◽  
Special Cases ◽  
The Neumann Problem ◽  
Reverse Hölder

Let [Formula: see text], [Formula: see text] be a bounded (semi-)convex domain in [Formula: see text] and the non-negative potential [Formula: see text] belong to the reverse Hölder class [Formula: see text]. Assume that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the Muckenhoupt weight class on [Formula: see text], the boundary of [Formula: see text]. In this paper, the authors show that, for any [Formula: see text], the Neumann problem for the Schrödinger equation [Formula: see text] in [Formula: see text] with boundary data in (weighted) [Formula: see text] is uniquely solvable. The obtained results in this paper essentially improve the known results which are special cases of the results obtained by Shen [Indiana Univ. Math. J. 43 (1994) 143–176] and Tao and Wang [Canad. J. Math. 56 (2004) 655–672], via extending the range [Formula: see text] of [Formula: see text] into [Formula: see text].

scholarly journals On the regularity of the total variation minimizers

2019 ◽  
Vol 23 (01) ◽  
pp. 1950082
Author(s):  
Alessio Porretta
Keyword(s):  
Image Processing ◽  
Total Variation ◽  
Source Term ◽  
Global Regularity ◽  
Regular Domain ◽  
Unique Minimizer ◽  
Convex Domains ◽  
Lipschitz Norm ◽  
Dimension Formula ◽  
Regularity Results

We prove regularity results for the unique minimizer of the total variation functional, currently used in image processing analysis since the work by Rudin, Osher and Fatemi. In particular, we show that if the source term [Formula: see text] is locally (respectively, globally) Lipschitz, then the solution has the same regularity with local (respectively, global) Lipschitz norm estimated accordingly. The result is proved in any dimension and for any (regular) domain. So far we extend a similar result proved earlier by Caselles, Chambolle and Novaga for dimension [Formula: see text] and (in case of the global regularity) for convex domains.

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