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Full-Text Articles in Physical Sciences and Mathematics
Dirichlet Problems In Perforated Domains, Robert Righi
Dirichlet Problems In Perforated Domains, Robert Righi
Theses and Dissertations--Mathematics
We establish W1,p estimates for solutions uε to the Laplace equation with Dirichlet boundary conditions in a bounded C1 domain Ωε, η perforated by small holes in ℝd. The bounding constants will depend explicitly on epsilon and eta, where epsilon is the order of the minimal distance between holes, and eta denotes the ratio between the size of the holes and epsilon. The proof relies on a large-scale Lp estimate for ∇uε, whose proof is divided into two main parts. First, we show that solutions of an intermediate problem for a …
Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell
Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell
Theses and Dissertations--Mathematics
In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H1-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, …
Convergence Rates In Periodic Homogenization Of Systems Of Elasticity, Zhongwei Shen, Jinping Zhuge
Convergence Rates In Periodic Homogenization Of Systems Of Elasticity, Zhongwei Shen, Jinping Zhuge
Mathematics Faculty Publications
This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in L2 for the mixed boundary value problems with bounded measurable coefficients.
Homogenization Of Stokes Systems With Periodic Coefficients, Shu Gu
Homogenization Of Stokes Systems With Periodic Coefficients, Shu Gu
Theses and Dissertations--Mathematics
In this dissertation we study the quantitative theory in homogenization of Stokes systems. We study uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L∞ estimates for the pressure as well as Liouville property for solutions in ℝd. We are able to obtain the boundary W{1,p} estimates in a bounded C1 domain for any 1 < p < ∞. We also study the convergence rates in L2 and H1 of Dirichlet and Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients.
Convergence Rates And Hölder Estimates In Almost-Periodic Homogenization Of Elliptic Systems, Zhongwei Shen
Convergence Rates And Hölder Estimates In Almost-Periodic Homogenization Of Elliptic Systems, Zhongwei Shen
Mathematics Faculty Publications
For a family of second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the coefficients. The results are used to investigate the problem of convergence rates. We also establish uniform Hölder estimates for the Dirichlet problem in a bounded C1,α domain.
Homogenization Of Stokes Systems And Uniform Regularity Estimates, Shu Gu, Zhongwei Shen
Homogenization Of Stokes Systems And Uniform Regularity Estimates, Shu Gu, Zhongwei Shen
Mathematics Faculty Publications
This paper is concerned with uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L∞ estimates for the pressure as well as a Liouville property for solutions in ℝd. We also obtain the boundary W1,p estimates in a bounded C1 domain for any 1 < p < ∞.