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G-convergence and Homogenization of Viscoelastic Flows
Published in Robert P. Gilbert, Ana Vasilic, Sandra Klinge, Alex Panchenko, Klaus Hackl, Applications of Homogenization Theory to the Study of Mineralized Tissue, 2020
Robert P. Gilbert, Ana Vasilic, Sandra Klinge, Alex Panchenko, Klaus Hackl
As an example of the application of the corrector operator technique, let Aε be elliptic differential operators in divergence form. In a bounded Lipschitz domain Ω⊂Rn, consider a sequence of problems Aεuε≡div(aε∇u)=f,
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Published in G Nakamura, S Saitoh, J K Seo, M Yamamoto, Inverse problems and related topics, 2019
Let D be a bounded Lipschitz domain in ℝn. Let Γ(x) be the fundamental solution of the Laplacian Δ: () Γ(x−y)={12πln|x−y|,n=2,1(2−n)ωn|x−y|2−n,n≥3,
An existence theorem for an unbounded dam with leaky boundary conditions
Published in C Bandle, J Bemelmans, M Chipot, J Saint Jean Paulin, I Shafrir, Elliptic and parabolic problems, 2020
Let Ω be an unbounded locally Lipschitz domain in ℝn (n ≥ 3). Ω represents an unbounded porous medium. The boundary Γ of Ω is divided into three parts: an impervious part S1, a part in contact with air S2, and a part covered by fluid S3. We denote by S3,ii ∈ I the different connected components of S3. Assuming that the flow in Ω has reached a steady state we are concerned with finding the pressure p of the fluid and the part of porous medium where some flow occurs, i.e., the wet subset A of Ω. We suppose that Ω ⊂ ℝn-1 × (—∞, H), H ∈ ℝ.
On the fractional order hyperbolic equation with random coefficients
Published in Applicable Analysis, 2023
In this paper, we mainly address a strictly hyperbolic equation with Dirichlet boundary condition described as follows, Here, , is a second-order moment stochastic process, which measures the oscillation on the principal fractional order operator about the starting time 0. Indeed, when , this is the wave equation. While is the case of plate equation. is a bounded Lipschitz domain with smooth boundary Γ. Recall the spectral theory in [13], the Dirichlet Laplacian is a self-adjoint, positive definite operator with compact resolvent whose spectrum set is composed of discrete points with finite multiplicity, which are arranged in order as Moreover, there exists a complete orthonormal basis composed of eigenfunctions as in , that is to say, for each . Readers can refer to [11,13,15] and references therein for more details concerned with the spectrum analysis.
Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations
Published in Applicable Analysis, 2022
Gregor Gantner, Dirk Praetorius
Throughout, is a mesh of the boundary of the bounded Lipschitz domain in the following sense: is a finite set of compact Lipschitz domains on Γ, i.e. each element T has the form , where is a compact2 Lipschitz domain in and is bi-Lipschitz. covers Γ, i.e. .For all with , the intersection has -dimensional Hausdorff measure zero.
A simple method of reconstructing a point-like scatterer according to time-dependent acoustic wave propagation
Published in Inverse Problems in Science and Engineering, 2021
If D is a Lipschitz domain in , then, for the point , there exists a radius α and a map such that is a bijection. Moreover, and are both Lipschitz continuous functions and where .