Questions tagged [homogenization]
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8 questions
2 votes
0 answers
109 views
The relation between capacity and the Poincaré inequality
Let $N > 2$ and let $\omega, \, \Omega \subset \mathbb{R}^N$ be domains containing the origin. Define $\varepsilon \omega := \{\varepsilon x : x \in \omega\}$ for $\varepsilon > 0$. I am ...
1 vote
0 answers
91 views
On the strong convergence of generators and the corresponding semigroups
Let $(E,\|\cdot\|)$ be a separable Hilbert space on $\mathbb{R}$. We consider a sequence of strongly continuous semigroup $\{T^{(n)}\}_{n=1}^\infty$ on $E$ [of course, for $n \in \mathbb{N}$ and $t>...
- 807
0 votes
0 answers
179 views
On the convergence of operators and their spectra
We consider a sequence of operators $\{L_n\}_{n=1}^\infty$. Each operator $L_n$ is a densely defined (possibly unbounded) closed linear operator on a real Hilbert space $H_n.$ The domain of $L_n$ is ...
- 807
1 vote
0 answers
75 views
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 ...
5 votes
1 answer
820 views
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 = ...
- 51
1 vote
0 answers
98 views
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 $\...
- 51
9 votes
1 answer
241 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 ...
- 53.1k
7 votes
1 answer
601 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 ...
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