TY - JOUR

T1 - Convergence for elliptic equations in periodic perforated domains

AU - Yeh, Li-Ming

PY - 2013/10/1

Y1 - 2013/10/1

N2 - Convergence for the solutions of elliptic equations in periodic perforated domains is concerned. Let ε denote the size ratio of the holes of a periodic perforated domain to the whole domain. It is known that, by energy method, the gradient of the solutions of elliptic equations is bounded uniformly in ε in L2 space. Also, when ε approaches 0, the elliptic solutions converge to a solution of some simple homogenized elliptic equation. In this work, above results are extended to general W1,p space for p>1. More precisely, a uniform W1,p estimate in ε for p∈(1, ∞] and a W1,p convergence result for p∈(nn-2,∞] for the elliptic solutions in periodic perforated domains are derived. Here n is the dimension of the space domain. One also notes that the Lp norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in ε.

AB - Convergence for the solutions of elliptic equations in periodic perforated domains is concerned. Let ε denote the size ratio of the holes of a periodic perforated domain to the whole domain. It is known that, by energy method, the gradient of the solutions of elliptic equations is bounded uniformly in ε in L2 space. Also, when ε approaches 0, the elliptic solutions converge to a solution of some simple homogenized elliptic equation. In this work, above results are extended to general W1,p space for p>1. More precisely, a uniform W1,p estimate in ε for p∈(1, ∞] and a W1,p convergence result for p∈(nn-2,∞] for the elliptic solutions in periodic perforated domains are derived. Here n is the dimension of the space domain. One also notes that the Lp norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in ε.

KW - Homogenized elliptic equation

KW - Periodic perforated domain

UR - http://www.scopus.com/inward/record.url?scp=84880509130&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2013.05.023

DO - 10.1016/j.jde.2013.05.023

M3 - Article

AN - SCOPUS:84880509130

VL - 255

SP - 1734

EP - 1783

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 7

ER -