Global boundedness and Hölder continuity of quasiminimizers with the general nonstandard growth conditions

2019 ◽  
Vol 185 ◽  
pp. 170-192
Author(s):  
Sungchol Kim ◽  
Dukman Ri
Keyword(s):  
Hölder Continuity ◽  
Growth Conditions ◽  
Holder Continuity ◽  
Nonstandard Growth ◽  
Nonstandard Growth Conditions ◽  
Global Boundedness

\mathfrak{B}_{1} classes of De Giorgi, Ladyzhenskaya, and Ural'tseva and their application to elliptic and parabolic equations with nonstandard growth

2019 ◽  
Vol 16 (3) ◽  
pp. 403-447
Author(s):  
Igor Skrypnik ◽  
Mykhailo Voitovych
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Hölder Continuity ◽  
Growth Conditions ◽  
Holder Continuity ◽  
Nonstandard Growth ◽  
Elliptic And Parabolic Equations ◽  
Functional Classes ◽  
Nonstandard Growth Conditions ◽  
Quasilinear Elliptic

The article provides an application of the generalized De Giorgi functional classes to the proof of the Hölder continuity of weak solutions to quasilinear elliptic and parabolic equations with nonstandard growth conditions.

scholarly journals On Solutions of a Parabolic Equation with Nonstandard Growth Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Growth Condition ◽  
Boundary Value ◽  
Growth Conditions ◽  
Boundary Value Condition ◽  
Nonstandard Growth ◽  
Partial Boundary Value Condition ◽  
Nonstandard Growth Condition ◽  
Nonstandard Growth Conditions ◽  
The Stability

A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficient bx,t on the time variable t, and the partial boundary value condition based on a submanifold of ∂Ω×0,T. How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.

scholarly journals Variational inequalities for energy functionals with nonstandard growth conditions

1998 ◽  
Vol 3 (1-2) ◽  
pp. 41-64 ◽  
Author(s):  
Martin Fuchs ◽  
Li Gongbao
Keyword(s):  
Variational Inequalities ◽  
Obstacle Problem ◽  
Partial Regularity ◽  
Growth Conditions ◽  
Energy Functionals ◽  
Nonstandard Growth ◽  
Bounded Lipschitz Domain ◽  
Growth Properties ◽  
Nonstandard Growth Conditions ◽  
Special Case

We consider the obstacle problem{minimize????????I(u)=?OG(?u)dx??among functions??u:O?Rsuch?that???????u|?O=0??and??u=F??a.e.for a given functionF?C2(O¯),F|?O<0and a bounded Lipschitz domainOinRn. The growth properties of the convex integrandGare described in terms of aN-functionA:[0,8)?[0,8)withlimt?8¯A(t)t-2<8. Ifn=3, we prove, under certain assumptions onG,C1,8-partial regularity for the solution to the above obstacle problem. For the special case whereA(t)=tln(1+t)we obtainC1,a-partial regularity whenn=4. One of the main features of the paper is that we do not require any power growth ofG.

scholarly journals Poincaré inequalities and Neumann problems for the variable exponent setting

2021 ◽  
Vol 4 (5) ◽  
pp. 1-22
Author(s):  
David Cruz-Uribe ◽  
◽  
Michael Penrod ◽  
Scott Rodney ◽  
Keyword(s):  
Variable Exponent ◽  
Growth Conditions ◽  
Linear Elliptic Equation ◽  
Poincaré Inequalities ◽  
Neumann Problems ◽  
Variable Exponent Spaces ◽  
Nonstandard Growth ◽  
Poincare Inequalities ◽  
Nonstandard Growth Conditions ◽  
Weighted Poincaré Inequalities

<abstract><p>In an earlier paper, Cruz-Uribe, Rodney and Rosta proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $ p $-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $ {p(\cdot)} $-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.</p></abstract>

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