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.

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.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

Continuous solutions and approximating scheme for fractional Dirichlet problems on Lipschitz domains

2018 ◽  
Vol 149 (2) ◽  
pp. 533-560
Author(s):  
Patricio Felmer ◽  
Erwin Topp
Keyword(s):  
Dirichlet Problem ◽  
Approximation Scheme ◽  
Linear Operators ◽  
Approximation Procedure ◽  
Dirichlet Problems ◽  
Uniqueness Of Solutions ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Continuous Boundary ◽  
Extra Assumption

In this paper, we study the fractional Dirichlet problem with the homogeneous exterior data posed on a bounded domain with Lipschitz continuous boundary. Under an extra assumption on the domain, slightly weaker than the exterior ball condition, we are able to prove existence and uniqueness of solutions which are Hölder continuous on the boundary. In proving this result, we use appropriate barrier functions obtained by an approximation procedure based on a suitable family of zero-th order problems. This procedure, in turn, allows us to obtain an approximation scheme for the Dirichlet problem through an equicontinuous family of solutions of the approximating zero-th order problems on ${\bar \Omega}$. Both results are extended to an ample class of fully non-linear operators.

scholarly journals Reflection Method for Mathematical Modeling of Potential Fields in Multi-Layer Media

2019 ◽  
Vol 224 ◽  
pp. 01004
Author(s):  
Oleg Yaremko ◽  
Natalia Yaremko
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Field Potential ◽  
Computer Code ◽  
Potential Fields ◽  
Reflection Method ◽  
Dirichlet Problems ◽  
Thermal Fields ◽  
Dirichlet Model ◽  
Fourth Kind

In this paper, potential fields in areas with plane and circular symmetry have been studied. In this case, the field potential is defined as the sum of solutions of Dirichlet model boundary value problems. The reflection method is used for the modeling of stationary thermal fields in multilayer media. By applying the reflection method, we found analytical solutions of boundary value problems with boundary conditions of the fourth kind for the Laplace equations and developed new computational algorithms. The developed algorithms can be easily implemented and transformed into a computer code. First of all, these algorithms implement consistent solutions of the Dirichlet problems for the model domains that allows using libraries of subroutines. Secondly, they have high algorithmic efficiency. It has been shown that reflection method is identical to the method of transformation operators and proved that transformation operator can be decomposed into a series of successive reflections from the external and internal boundaries. Finally, a physical interpretation of the reflection method has been discussed in detail.

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