scholarly journals Boundary value problems for a class of elliptic operator pencils

2000 ◽  
Vol 38 (4) ◽  
pp. 410-436 ◽  
Author(s):  
R. Denk ◽  
R. Mennicken ◽  
L. Volevich
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Boundary Value ◽  
Operator Pencils

scholarly journals On Boundary Value Problems for Elliptic Equations in a Singular Domain

1969 ◽  
Vol 36 ◽  
pp. 99-115
Author(s):  
Kazunari Hayashida
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Tangent Line ◽  
Boundary Operators ◽  
One To One ◽  
Singular Domain

1. Let Ω be a bounded domain in the plane and denotes its closure and boundary by Ω̅ and ∂Ω, respectively. We shall say that the domain Ω is regular, if every point P ∈ ∂û has an 2-dimensional neighborhood U such that dΩ ∩ U can be mapped in a one-to-one way onto a portion of the tangent line through P by a mapping T which together with its inverse is infinitely differentiable. Let L be an elliptic operator of order 2m defined in Ω̅ and let be a normal set of boundary operators of orders mf <2m. If f is a given function in Ω, the boundary value problem II(L,f,Bj) will be to find a solution u ofsatisfyingBju = 0 on ∂Ω, j = 1, …, m.

scholarly journals Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains

1995 ◽  
Vol 125 (1) ◽  
pp. 99-114 ◽  
Author(s):  
Gianni Dal Maso ◽  
Annalisa Malusa
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Radon Measure ◽  
Boundary Value ◽  
Approximation Procedure ◽  
Dirichlet Problems ◽  
Perforated Domains ◽  
Positive Radon Measure

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.

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