Existence and Hölder continuity to solutions of the complex Monge–Ampère-type equations with measures vanishing on pluripolar subsets

2021 ◽  
Author(s):  
Le Mau Hai ◽  
Vu Van Quan
Keyword(s):  
Pseudoconvex Domain ◽  
Hölder Continuity ◽  
Type Equation ◽  
Holder Continuity ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Strictly Pseudoconvex Domain ◽  
Monge Ampere

In this paper, we establish existence of Hölder continuous solutions to the complex Monge–Ampère-type equation with measures vanishing on pluripolar subsets of a bounded strictly pseudoconvex domain [Formula: see text] in [Formula: see text].

On nonhomeomorphic mappings with the inverse Poletsky inequality

2020 ◽  
Vol 17 (3) ◽  
pp. 414-436
Author(s):  
Evgeny Sevost'yanov ◽  
Serhii Skvortsov ◽  
Oleksandr Dovhopiatyi
Keyword(s):  
Boundary Behavior ◽  
Hölder Continuity ◽  
Continuous Extension ◽  
Quasiconformal Mappings ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Finite Distortion ◽  
Mappings Of Finite Distortion ◽  
Lipschitz Continuous ◽  
Inverse Inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

Partial Hölder continuity of minimizers of functionals satisfying a VMO condition

2017 ◽  
Vol 10 (1) ◽  
pp. 83-110 ◽  
Author(s):  
Christopher S. Goodrich
Keyword(s):  
Large Class ◽  
Partial Regularity ◽  
Hölder Continuity ◽  
Full Measure ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Open Set ◽  
Principal Assumption ◽  
Class Of Functions

AbstractFor a bounded, open set${\Omega\hskip-0.569055pt\subseteq\hskip-0.569055pt\mathbb{R}^{n}}$we consider the partial regularity of vectorial minimizers${u\hskip-0.853583pt:\hskip-0.853583pt\Omega\hskip-0.853583pt\rightarrow\hskip-% 0.853583pt\mathbb{R}^{N}}$of the functional$u\mapsto\int_{\Omega}f(x,u,Du)\,dx,$where${f:\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times n}\rightarrow\mathbb{R}}$. The principal assumption we make is thatfis asymptotically related to a function of the form${(x,u,\xi)\mapsto a(x,u)F(\xi)}$, whereFpossessesp-Uhlenbeck structure and the partial maps${x\mapsto a(x,\cdot\,)}$and${u\mapsto a(\,\cdot\,,u)}$are, respectively, of class VMO and${\mathcal{C}^{0}}$. We demonstrate that any minimizer${u\in W^{1,p}(\Omega)}$of this functional is Hölder continuous on an open set${\Omega_{0}}$of full measure. Finally, we show by means of an example that our asymptotic relatedness condition is very general and permits a large class of functions.

scholarly journals Hölder continuous solutions to Monge–Ampère equations

2014 ◽  
Vol 16 (4) ◽  
pp. 619-647 ◽  
Author(s):  
Jean-Pierre Demailly ◽  
Sławomir Dinew ◽  
Vincent Guedj ◽  
Pham Hoang Hiep ◽  
Sławomir Kołodziej ◽  
...  
Keyword(s):  
Continuous Solutions ◽  
Hölder Continuous ◽  
Monge Ampere

scholarly journals Hölder continuous solutions to Monge-Ampère equations

2008 ◽  
Vol 40 (6) ◽  
pp. 1070-1080 ◽  
Author(s):  
V. Guedj ◽  
S. Kolodziej ◽  
A. Zeriahi
Keyword(s):  
Continuous Solutions ◽  
Hölder Continuous ◽  
Monge Ampere

scholarly journals Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Sławomir Kołodziej ◽  
Ngoc Cuong Nguyen
Keyword(s):  
Hermitian Manifold ◽  
General Measure ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Hermitian Manifolds ◽  
Right Hand ◽  
The Right ◽  
Monge Ampere ◽  
Compact Hermitian Manifold

AbstractWe prove the existence of a continuous quasi-plurisubharmonic solution to the Monge–Ampère equation on a compact Hermitian manifold for a very general measure on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh–Nguyen–Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh–Nguyen.

scholarly journals Pointwise regularity for solutions of double obstacle problems on metric spaces

2011 ◽  
Vol 109 (2) ◽  
pp. 185 ◽  
Author(s):  
Zohra Farnana
Keyword(s):  
Metric Spaces ◽  
Hölder Continuity ◽  
Obstacle Problems ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Pointwise Regularity

We study continuity at a given point for solutions of double obstacle problems. We obtain pointwise continuity of the solutions for discontinuous obstacles. We also show Hölder continuity for solutions of the double obstacle problems if the obstacles are Hölder continuous.

scholarly journals Hölder continuity of Oseledets splittings for semi-invertible operator cocycles

2016 ◽  
Vol 38 (3) ◽  
pp. 961-981 ◽  
Author(s):  
DAVOR DRAGIČEVIĆ ◽  
GARY FROYLAND
Keyword(s):  
Hilbert Space ◽  
Lyapunov Exponents ◽  
Hölder Continuity ◽  
Compact Operators ◽  
Invertible Operator ◽  
Compact Sets ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Large Measure ◽  
Space Of Compact Operators

For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo et al [On Hölder-continuity of Oseledets subspaces J. Lond. Math. Soc.93 (2016) 194–218] by considering possibly non-invertible cocycles, which, in addition, may take values in the space of compact operators on a Hilbert space. As a by-product of our work, we also show that a non-invertible cocycle with non-vanishing Lyapunov exponents exhibits non-uniformly hyperbolic behaviour (in the sense of Pesin) on a set of full measure.

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