Questions tagged [homogenization]

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Betti numbers of a polynomial ideal after homogenization

Let $I \subset k[x_1,\ldots,x_n]$ be an ideal in a polinomial ring over a field $k$. Assume that $I$ is quasihomogeneous (that is, $I$ is not homogeneous with the usual grading, but it is homogeneous ...
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Eigenvalue and eigenfunction convergence

Consider a bounded Euclidean domain $\Omega \subset \mathbb{R}^n$ (for simplicity, let's say, $\Omega$ has smooth boundary and is simply connected). Let $p \in \Omega$ be a point, and call $\Omega_n = ...
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Density gradient of Navier-Stokes equations in perforated domain

Question: Is it possible to bound the density gradient $\nabla \rho_\epsilon$ of strong solutions to the compressible NSE uniformly in $L^\gamma$? Preliminaries: Consider a bounded connected domain $\...
9
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186 views

Geometry of the positive definite cone, versus homogenization of elliptic PDEs

Homogenization is a process that assigns to a positive definite-valued map $x\mapsto S(x)$ a non-trivial but physically meaningful average $\bar S$. There are various settings, for instance stochastic ...
7
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1answer
418 views

Dependence of solutions on parameters in partial differential equations

In the standard homogenization problem $$-\nabla.\left(A\left(x,\frac{x}{\epsilon}\right)\nabla u^{\epsilon}(x)\right)=f\ \mbox{in } \Omega,$$ the homogenized matrix $A_0$ is given in terms of ...