$$L^{\Phi }-L^{\infty }$$ L Φ - L ∞ inequalities and new remarks on the Hölder continuity of the quasi-minima of scalar integral functionals with general growths

2015 ◽  
Vol 22 (1) ◽  
pp. 165-212 ◽  
Author(s):  
Tiziano Granucci
Keyword(s):  
Hölder Continuity ◽  
Integral Functionals ◽  
Holder Continuity ◽  
Scalar Integral

scholarly journals On the Hölder continuity for a class of vectorial problems

2019 ◽  
Vol 9 (1) ◽  
pp. 1008-1025
Author(s):  
Giovanni Cupini ◽  
Matteo Focardi ◽  
Francesco Leonetti ◽  
Elvira Mascolo
Keyword(s):  
Final Section ◽  
Hölder Continuity ◽  
Integral Functionals ◽  
Holder Continuity ◽  
Local Minimizers ◽  
Regularity Of Minimizers ◽  
Rank One ◽  
Suitable Structure ◽  
Local Hölder Continuity ◽  
Special Classes

Abstract In this paper we prove local Hölder continuity of vectorial local minimizers of special classes of integral functionals with rank-one and polyconvex integrands. The energy densities satisfy suitable structure assumptions and may have neither radial nor quasi-diagonal structure. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the Hölder continuity. In the final section, we provide some non-trivial applications of our results.

Partial and full boundary regularity for non-autonomous functionals with Φ-growth conditions

2019 ◽  
Vol 31 (4) ◽  
pp. 1027-1050
Author(s):  
Flavia Giannetti ◽  
Antonia Passarelli di Napoli ◽  
Atsushi Tachikawa
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Hölder Continuity ◽  
Boundary Regularity ◽  
Growth Conditions ◽  
Integral Functionals ◽  
Holder Continuity ◽  
Alpha Beta

Abstract We prove partial and full boundary Hölder continuity, under a suitable regularity on the boundary datum, of the minimizers of non-autonomous integral functionals of the type \int_{\Omega}\Phi\bigl{(}(A^{\alpha\beta}_{ij}(x,u)D_{i}u^{\alpha}D_{j}u^{% \beta})^{\frac{1}{2}}\bigr{)}\mathop{}\!dx, where {\Omega\subset\mathbb{R}^{n}} is a bounded domain, {\Phi(t)=t^{p}\log^{\alpha}(e+t)} with {1<p\leq n} and {\alpha>0} , and {A(x,s)=(A^{\alpha\beta}_{ij}(x,s))} is a uniformly elliptic, bounded and continuous function.

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

scholarly journals Hölder continuity of weak solutions of a class of linear equations with boundary degeneracy

2007 ◽  
Vol 239 (1) ◽  
pp. 99-131 ◽  
Author(s):  
Chunpeng Wang ◽  
Lihe Wang ◽  
Jingxue Yin ◽  
Shulin Zhou
Keyword(s):  
Weak Solutions ◽  
Hölder Continuity ◽  
Linear Equations ◽  
Holder Continuity ◽  
Boundary Degeneracy

scholarly journals Regularity of the m-symphonic map

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Masashi Misawa ◽  
Nobumitsu Nakauchi
Keyword(s):  
Critical Points ◽  
Conformal Invariance ◽  
Hölder Continuity ◽  
Energy Functional ◽  
Higher Dimension ◽  
Holder Continuity ◽  
New Energy ◽  
Symphonic Map

AbstractWe introduce a new energy functional of conformal invariance and consider its critical points, named the m-symphonic map. We study a Hölder continuity of m-symphonic maps from domains of $$\mathbb {R}^m$$ R m into the spheres in the higher dimension $$m \ge 4$$ m ≥ 4 .

-Hölder linearization of hyperbolic diffeomorphisms with resonance

2014 ◽  
Vol 36 (1) ◽  
pp. 310-334 ◽  
Author(s):  
WENMENG ZHANG ◽  
WEINIAN ZHANG
Keyword(s):  
Hölder Continuity ◽  
Three Dimensional ◽  
Holder Continuity ◽  
Hyperbolic Diffeomorphisms ◽  
Analytic Mapping

Concerning hyperbolic diffeomorphisms, one expects a better smoothness of linearization, but it may be confined by resonance among eigenvalues. Hartman gave a three-dimensional analytic mapping with resonance which cannot be linearized by a Lipschitz conjugacy. Since then, efforts have been made to give the ${\it\alpha}$-Hölder continuity of the conjugacy and hope the exponent ${\it\alpha}<1$ can be as large as possible. Recently, it was proved for some weakly resonant hyperbolic diffeomorphisms that ${\it\alpha}$ can be as large as we expect. In this paper we prove that this result holds for all $C^{\infty }$ weakly resonant hyperbolic diffeomorphisms.

Sign in / Sign up

Export Citation Format

Share Document