scholarly journals On Liouville theorem and Hölder continuity of weak solutions to some quasilinear elliptic systems of higher order

1992 ◽  
Vol 117 (4) ◽  
pp. 373-392
Author(s):  
Lubomír Balanda ◽  
Eugen Viszus
Keyword(s):  
Weak Solutions ◽  
Liouville Theorem ◽  
Hölder Continuity ◽  
Elliptic Systems ◽  
Higher Order ◽  
Holder Continuity ◽  
Quasilinear Elliptic Systems ◽  
Quasilinear Elliptic

\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 Partial Hölder continuity for nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group

2018 ◽  
Vol 7 (1) ◽  
pp. 97-116 ◽  
Author(s):  
Jialin Wang ◽  
Juan J. Manfredi
Keyword(s):  
Heisenberg Group ◽  
Weak Solutions ◽  
Hölder Continuity ◽  
Harmonic Approximation ◽  
Elliptic Systems ◽  
Natural Growth ◽  
Holder Continuity ◽  
Model Case ◽  
Horizontal Gradients ◽  
Vmo Coefficients

AbstractWe consider nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group and prove partial Hölder continuity results for weak solutions using a generalization of the technique of {\mathcal{A}}-harmonic approximation. The model case is the following non-degenerate p-sub-Laplace system with super-quadratic natural growth with respect to the horizontal gradients Xu:-\sum_{i=1}^{2n}X_{i}\bigl{(}a(\xi\/)(1+|Xu|^{2})^{{(p-2)/2}}X_{i}u^{\alpha}% \bigr{)}=f^{\alpha},\quad\alpha=1,2,\ldots,N,where {a(\xi\/)\in\mathrm{VMO}} and {2<p<\infty}.

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