scholarly journals A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group

2019 ◽  
Vol 18 (3) ◽  
pp. 1091-1115
Author(s):  
Pablo Ochoa ◽  
◽  
Julio Alejo Ruiz
Keyword(s):  
Heisenberg Group ◽  
Weak Solutions ◽  
Quasilinear Equations

scholarly journals A Hopf-type Boundary Point Lemma for Pairs of Solutions to Quasilinear Equations

2019 ◽  
Vol 62 (3) ◽  
pp. 607-621 ◽  
Author(s):  
Leobardo Rosales
Keyword(s):  
Weak Solutions ◽  
Boundary Point ◽  
Quasilinear Equation ◽  
Divergence Form ◽  
Quasilinear Equations ◽  
Differentiable Functions ◽  
The Difference ◽  
Type Boundary

AbstractWe present a Hopf boundary point lemma for the difference between two Hölder continuously differentiable functions, each weak solutions to a divergence-form quasilinear equation, under mild boundedness assumptions on the coefficients of this equation.

scholarly journals On the rate of the volume growth for symmetric viscous heat-conducting gas flows with a free boundary

2006 ◽  
Vol 2006 ◽  
pp. 1-8
Author(s):  
Alexander Zlotnik
Keyword(s):  
Free Boundary ◽  
Weak Solutions ◽  
Volume Growth ◽  
External Boundary ◽  
Gas Flows ◽  
Positive Density ◽  
Heat Conducting ◽  
Quasilinear Equations ◽  
Viscous Heat ◽  
Gas Volume

The system of quasilinear equations for symmetric flows of a viscous heat-conducting gas with a free external boundary is considered. For global in time weak solutions having nonstrictly positive density, the linear in time two-sided bounds for the gas volume growth are established.

A result of multiplicity of solutions for a class of quasilinear equations

2012 ◽  
Vol 55 (2) ◽  
pp. 291-309 ◽  
Author(s):  
Claudianor O. Alves ◽  
Giovany M. Figueiredo ◽  
Uberlandio B. Severo
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Category Theory ◽  
Nonlinear Term ◽  
Positive Parameter ◽  
Quasilinear Equations ◽  
Subcritical Growth ◽  
Minimax Methods ◽  
Multiplicity Of Positive Solutions

AbstractWe establish the multiplicity of positive weak solutions for the quasilinear Dirichlet problem−Lpu+ |u|p−2u=h(u)in Ωλ,u= 0 on ∂Ωλ, where Ωλ= λΩ, Ω is a bounded domain in ℝN, λ is a positive parameter,Lpu≐ Δpu+ Δp(u2)uand the nonlinear termh(u) has subcritical growth. We use minimax methods together with the Lyusternik–Schnirelmann category theory to get multiplicity of positive solutions.

scholarly journals Existence and multiplicity results for quasilinear equations in the Heisenberg group

2019 ◽  
Vol 39 (2) ◽  
pp. 247-257 ◽  
Author(s):  
Patrizia Pucci
Keyword(s):  
Heisenberg Group ◽  
Quasilinear Equation ◽  
Divergence Form ◽  
Infinitely Many Solutions ◽  
Entire Solutions ◽  
Quasilinear Equations ◽  
Multiplicity Results ◽  
Minimax Theory ◽  
Existence And Multiplicity ◽  
General Elliptic

In this paper we complete the study started in [Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl. 4 (2019)] on entire solutions for a quasilinear equation \((\mathcal{E}_{\lambda})\) in \(\mathbb{H}^{n}\), depending on a real parameter \(\lambda\), which involves a general elliptic operator \(\mathbf{A}\) in divergence form and two main nonlinearities. Here, in the so called sublinear case, we prove existence for all \(\lambda\gt 0\) and, for special elliptic operators \(\mathbf{A}\), existence of infinitely many solutions \((u_k)_k\).

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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