Homogenization of the Poisson Equation in a Perforated Domain with the Signorini Condition and the Third Boundary Condition on the Cavity Boundaries

This paper deals with the problemΔu=gonGand∂u/∂n+uf=Lon∂G. Here,G⊂ℝm,m>2, is a bounded domain with Lyapunov boundary,fis a bounded nonnegative function on the boundary ofG,Lis a bounded linear functional onW1,2(G)representable by a real measureμon the boundary ofG, andg∈L2(G)∩Lp(G),p>m/2. It is shown that a weak solution of this problem is bounded inGif and only if the Newtonian potential corresponding to the boundary conditionμis bounded inG.

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REINFORCEMENT OF THE POISSON EQUATION BY A THIN LAYER

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202511005283 ◽

2011 ◽

Vol 21 (05) ◽

pp. 1153-1192 ◽

Cited By ~ 5

Author(s):

JINGYU LI ◽

KAIJUN ZHANG

Keyword(s):

Boundary Condition ◽

Shear Modulus ◽

Poisson Equation ◽

Dirichlet Boundary Condition ◽

Dirichlet Boundary ◽

Neumann Boundary ◽

Robin Boundary Condition ◽

Oscillation Period ◽

The Poisson Equation ◽

Robin Boundary

We consider the problem of reinforcing an elastic medium by a strong, rough, thin external layer. This model is governed by the Poisson equation with homogeneous Dirichlet boundary condition. We characterize the asymptotic behavior of the solution as the shear modulus of the layer goes to infinity. We find that there are four types of behaviors: the limiting solution satisfies Poisson equation with Dirichlet boundary condition, Robin boundary condition or Neumann boundary condition, or the limiting solution does not exist. The specific type depends on the integral of the load on the medium, the curvature of the interface and the scaling relations among the shear modulus, the thickness and the oscillation period of the layer.

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Homogenization of the Poisson equation in a non-periodically perforated domain

Asymptotic Analysis ◽

10.3233/asy-201667 ◽

2021 ◽

pp. 1-27

Author(s):

Xavier Blanc ◽

Sylvain Wolf

Keyword(s):

Poisson Equation ◽

Dirichlet Boundary ◽

Rocky Mountain ◽

Dirichlet Boundary Conditions ◽

Perforated Domain ◽

Periodic Case ◽

Localized Deformation ◽

The Poisson Equation ◽

Homogeneous Dirichlet Boundary Conditions ◽

Periodic Setting

We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by ε > 0, and is proportional to the distance between neighbouring perforations. In the periodic case, the homogenized problem (obtained in the limit ε → 0) is well understood (see (Rocky Mountain J. Math. 10 (1980) 125–140)). We extend these results to a non-periodic case which is defined as a localized deformation of the periodic setting. We propose geometric assumptions that make precise this setting, and we prove results which extend those of the periodic case: existence of a corrector, convergence to the homogenized problem, and two-scale expansion.

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Local Solvability of a Linear System with a Fractional Derivative in Time in a Boundary Condition

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2015-0058 ◽

2015 ◽

Vol 18 (4) ◽

Cited By ~ 2

Author(s):

Nataliya Vasylyeva

Keyword(s):

Boundary Condition ◽

Linear System ◽

Fractional Derivative ◽

Classical Solution ◽

Poisson Equation ◽

Existence And Uniqueness ◽

Local Solvability ◽

The Poisson Equation ◽

The Right ◽

Coercive Estimates

AbstractIn this paper we analyze a linear system for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides depended on time. First, we prove existence and uniqueness of the classical solution to this problem, and provide the coercive estimates of the solution. Second, based on the obtained results we establish one-to-one solvability to a linear system of a general form in the H¨older spaces.

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