scholarly journals Boundary conditions for the Stokes fluid in a bounded domain with a thin layer

2016 ◽  
Vol 9 (4) ◽  
pp. 797-812 ◽  
Author(s):  
Hongxing Zhao ◽  
Zhengan Yao
Keyword(s):  
Boundary Conditions ◽  
Thin Layer ◽  
Bounded Domain ◽  
Stokes Fluid

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

Analysis of an elliptic system with infinitely many solutions

2017 ◽  
Vol 6 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Carmen Cortázar ◽  
Manuel Elgueta ◽  
Jorge García-Melián
Keyword(s):  
Boundary Conditions ◽  
Elliptic System ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Infinitely Many Solutions ◽  
Smooth Bounded Domain ◽  
Content Type ◽  
Outward Unit ◽  
Unit Normal

AbstractWe consider the elliptic system ${\Delta u\hskip-0.284528pt=\hskip-0.284528ptu^{p}v^{q}}$, ${\Delta v\hskip-0.284528pt=\hskip-0.284528ptu^{r}v^{s}}$ in Ω with the boundary conditions ${{\partial u/\partial\eta}=\lambda u}$, ${{\partial v/\partial\eta}=\mu v}$ on ${\partial\Omega}$, where Ω is a smooth bounded domain of ${\mathbb{R}^{N}}$, ${p,s>1}$, ${q,r>0}$, ${\lambda,\mu>0}$ and η stands for the outward unit normal. Assuming the “criticality” hypothesis ${(p-1)(s-1)=qr}$, we completely analyze the values of ${\lambda,\mu}$ for which there exist positive solutions and give a detailed description of the set of solutions.

Study of Stokes dynamical system in a thin domain with Fourier and Tresca boundary conditions

2019 ◽  
pp. 2150007 ◽  
Author(s):  
Y. Letoufa ◽  
H. Benseridi ◽  
M. Dilmi
Keyword(s):  
Boundary Conditions ◽  
Variational Formulation ◽  
Reynolds Equation ◽  
Mixed Boundary ◽  
Three Dimensional ◽  
Mixed Boundary Conditions ◽  
Dynamic Regime ◽  
Thin Domain ◽  
Problem Statement ◽  
Stokes Fluid

Asymptotic analysis of an incompressible Stokes fluid in a dynamic regime in a three-dimensional thin domain [Formula: see text] with mixed boundary conditions and Tresca friction law is studied in this paper. The problem statement and variational formulation of the problem are reformulated in a fixed domain. In which case, the estimates on velocity and pressure are proved. These estimates will be useful in order to give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.

scholarly journals A biharmonic equation with singular nonlinearity

2011 ◽  
Vol 55 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
The Green Function ◽  
Uniqueness Of A Solution

AbstractWe study the biharmonic equation Δ2u=u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn,n≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

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