Asymptotic Behaviour for Dirichlet Problems in Domains Bounded by Thin Layers

1989 ◽  
pp. 193-249 ◽  
Author(s):  
Giuseppe Buttazzo ◽  
Gianni Dal Maso ◽  
Umberto Mosco
Keyword(s):  
Asymptotic Behaviour ◽  
Thin Layers ◽  
Dirichlet Problems

Stochastic reinforcement problems and ergodic theory

1992 ◽  
Vol 121 (1-2) ◽  
pp. 33-53 ◽  
Author(s):  
M. Balzano ◽  
G. Paderni
Keyword(s):  
Asymptotic Behaviour ◽  
Ergodic Theory ◽  
Random Function ◽  
Limit Problem ◽  
Thin Layers ◽  
Ergodic Theorems ◽  
Dirichlet Problems

SynopsisWe study the asymptotic behaviour of Dirichlet problems in domains of R2 bounded by thin layers whose thickness is given by means of an assigned ergodic random function. Using a capacitary method together with ergodic theorems for additive and superadditive processes, we are able to characterise the limit problem precisely.

Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

2020 ◽  
pp. 1-19
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behaviour ◽  
Unique Solution ◽  
Blow Up ◽  
Critical Values ◽  
Smooth Domain ◽  
Dirichlet Problems ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝ n , g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$ , b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

Non-trivial solutions of elliptic equations at resonance

1980 ◽  
Vol 85 (1-2) ◽  
pp. 119-129 ◽  
Author(s):  
Klaus Thews
Keyword(s):  
Asymptotic Behaviour ◽  
Elliptic Equations ◽  
The Other ◽  
Infinitely Many Solutions ◽  
Dirichlet Problems ◽  
At Resonance ◽  
Other Hand

SynopsisIn this paper we show that Dirichlet problems at resonance, being of the type −Δu(x) = λku(x) + g(u(x)), x є G, u(x) = 0 for x є ∂G, g(−u) = −g(u), admit multiple non-trivial solutions provided the non-linearity interacts in some sense with the spectrum of −Δ. In contrast to other work on this subject we deal with the case that g(u) is very small for large arguments, for instance g(u) = 0 for |u| large. On the other hand if and g satisfies a certain concavity condition at 0 the existence of infinitely many solutions is shown independent of the asymptotic behaviour of g.

Homogenization of singularly perturbed Dirichlet problems in perforated domains

2000 ◽  
Vol 130 (1) ◽  
pp. 35-51
Author(s):  
Viêt Há Hoáng
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behaviour ◽  
Error Estimates ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Perforated Domain ◽  
Dirichlet Problems ◽  
Perforated Domains

We study the singularly perturbed problem —εαΔuε + uε = f (α > 0) with the Dirichlet boundary condition in a perforated domain, in which the holes are distributed periodically with period 2ε throughout a fixed domain Ω. The asymptotic behaviour of uε when ε → 0, together with corrector results and error estimates in L2(Ω), are deduced for all sizes of holes. The behaviour of uε in is obtained for the cases where the size of holes is of order ε or is of a sufficiently smaller order. When the holes' size is of a sufficiently small order, as expected, uε has similar behaviour to that in the case of a non-varying domain.

scholarly journals Asymptotic behaviour of a Bingham fluid in thin layers

2004 ◽  
Vol 293 (2) ◽  
pp. 405-418 ◽  
Author(s):  
Renata Bunoiu ◽  
Srinivasan Kesavan
Keyword(s):  
Asymptotic Behaviour ◽  
Bingham Fluid ◽  
Thin Layers

Homogenisation of Dirichlet problems for monotone operators in varying domains

1997 ◽  
Vol 127 (3) ◽  
pp. 457-478 ◽  
Author(s):  
Juan Casado-Díaz
Keyword(s):  
Asymptotic Behaviour ◽  
Monotone Operator ◽  
Monotone Operators ◽  
Limit Problem ◽  
Dirichlet Problems

We study the asymptotic behaviour, for a sequence of varying open sets Ωn, of the solutionsunof nonlinear Dirichlet problems for a monotone Leray–Lions operator. The method is based on the comparison between the gradient ofunand the corrector for thep-Laplacian corresponding to the same geometry as the monotone operator. The representation of the limit problem and a corrector result are obtained.

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