A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term

Representation formulas for $L^\infty$ norms of weakly convergent sequences of gradient fields in homogenization

We examine the composition of the $L^{\infty}$ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the $L^{\infty}$ norms of gradient fields. The local representation formulas are expressed in terms of the weak limit of the …

Closure and commutability results for Gamma-limits and the geometric linearization and homogenization of multi-well energy functionals

Under a suitable notion of equivalence of integral densities we prove a $Γ$-closure theorem for integral functionals: The limit of a sequence of $Γ$-convergent families of such functionals is again a $Γ$-convergent family. Its $Γ$-limit is the limit of the $Γ$-limits of the original problems. This result not only provides a common basic principle for a number of linearization and homogenization results in elasticity theory. It also allows for new applications as we exemplify by proving that geometric linearization and homogenization of multi-well energy functionals commute.

A rational framework for dynamic homogenization at finite wavelengths and frequencies

In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (with or without the source term) at finite frequencies and wavelengths spanning the first Brioullin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency (FW-FF) homogenization is pursued in~$\mathbb{R}^d$ via second-order asymptotic expansion about the apex…

Hamilton-Jacobi方程准周期均匀化的多项式收敛速率及其在遍历估计中的应用

Polynomial Convergence Rate for Quasi-Periodic Homogenization of Hamilton-Jacobi Equations and Application to Ergodic Estimates

本文利用具Diophantine频率有界准周期函数的新定量遍历估计，证明了具准周期势的Hamilton-Jacobi方程均匀化的多项式收敛速率，建立了收敛速率与有效Hamiltonian正则性的联系，并应用于研究无界准周期函数Birkhoff平均的收敛速率。

Homogenization of degenerate coupled transport processes in porous media with memory terms

In this paper we establish a homogenization result for a doubly nonlinear parabolic system arising from the hygro-thermo-chemical processes in porous media taking into account memory phenomena. We present a meso-scale model of the composite (heterogeneous) material where each component is considered as a porous system and the voids of the skeleton are partially saturated with liquid water. It is shown that the solution of the meso-scale problem is two-scale convergent to that of the upscaled problem as the spatial parameter goes to zero.

On the spectral asymptotics of waves in periodic media with Dirichlet or Neumann exclusions

We consider homogenization of the scalar wave equation in periodic media at finite wavenumbers and frequencies, with the focus on continua characterized by: (a) arbitrary Bravais lattice in $\mathbb{R}^d$, $d\!\geqslant\!2$, and (b) exclusions i.e. "voids" that are subject to homogenous (Neumann or Dirichlet) boundary conditions. Making use of the Bloch wave expansion, we pursue this goal via asymptotic ansatz featuring the "spectral distance" from a given wavenumber-eigenfrequency pair (within the first Brillouin zone) as the perturbation parameter. We then introduce the effective wave motion…