Hölder continuity and Harnack inequality for De Giorgi classes related to Hörmander vector fields

1995 ◽  
Vol 168 (1) ◽  
pp. 171-188 ◽  
Author(s):  
Silvana Marchi
Keyword(s):  
Harnack Inequality ◽  
Vector Fields ◽  
Hölder Continuity ◽  
Holder Continuity ◽  
Hörmander Vector Fields

scholarly journals Remarks on parabolic De Giorgi classes

2021 ◽  
Author(s):  
Naian Liao
Keyword(s):  
Harnack Inequality ◽  
Hölder Continuity ◽  
Holder Continuity ◽  
Type Estimate ◽  
Local Boundedness ◽  
Heavy Machinery ◽  
Weak Harnack Inequality ◽  
Logarithmic Type ◽  
Local Hölder Continuity

AbstractWe make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser’s ideas, thus avoiding De Giorgi’s heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, without any covering argument.

scholarly journals Harnack inequality for a class of functionals with non-standard growth via De Giorgi’s method

2018 ◽  
Vol 7 (2) ◽  
pp. 167-182 ◽  
Author(s):  
Jihoon Ok
Keyword(s):  
Harnack Inequality ◽  
Hölder Continuity ◽  
Regularity Theory ◽  
Holder Continuity ◽  
Local Boundedness

AbstractWe study the regularity theory of quasi-minimizers of functionals with {L^{p(\,\cdot\,)}\log L}-growth. In particular, we prove the Harnack inequality and, in addition, the local boundedness and the Hölder continuity of the quasi-minimizers. We directly prove our results via De Giorgi’s method.

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.

A novel approach to Hölder continuity of a class of parametric variational–hemivariational inequalities

2021 ◽  
Vol 49 (2) ◽  
pp. 283-289
Author(s):  
Nguyen Van Hung ◽  
Vo Minh Tam ◽  
Zhenhai Liu ◽  
Jen Chih Yao
Keyword(s):  
Hölder Continuity ◽  
Hemivariational Inequalities ◽  
Holder Continuity ◽  
Novel Approach

scholarly journals Linearization and Hölder continuity for nonautonomous systems

2021 ◽  
Vol 297 ◽  
pp. 536-574
Author(s):  
Lucas Backes ◽  
Davor Dragičević ◽  
Kenneth J. Palmer
Keyword(s):  
Hölder Continuity ◽  
Holder Continuity ◽  
Nonautonomous Systems
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