English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We consider the operators \[  \nabla_X\cdot(A(X)\nabla_X),\ \nabla_X\cdot(A(X)\nabla_X)-\partial_t,\\nabla_X\cdot(A(X)\nabla_X)+X\cdot\nabla_Y-\partial_t, \] where $X\in \Omega$,$(X,t)\in \Omega\times \mathbb R$ and $(X,Y,t)\in \Omega\times \mathbbR^m\times \mathbb R$, respectively, and where $\Omega\subset\mathbb R^m$ is a(unbounded) Lipschitz domain with defining function $\psi:\mathbbR^{m-1}\to\mathbb R$ being Lipschitz with constant bounded by $M$. Assume thatthe elliptic measure associated to the first of these operators is mutuallyabsolutely continuous with respect to the surface measure $\mathrm{d}\sigma(X)$, and that the corresponding Radon-Nikodym derivative or Poissonkernel satisfies a scale invariant reverse H\"older inequality in $L^p$, forsome fixed $p$, $1<p<\infty$, with constants depending only on the constants of$A$, $m$ and the Lipschitz constant of $\psi$, $M$. Under this assumption weprove that then the same conclusions are also true for the parabolic measuresassociated to the second and third operator with $\mathrm{d} \sigma(X)$replaced by the surface measures $\mathrm{d} \sigma(X)\mathrm{d} t$ and$\mathrm{d} \sigma(X)\mathrm{d} Y\mathrm{d} t$, respectively. This structuraltheorem allows us to reprove several results previously established in theliterature as well as to deduce new results in, for example, the context ofhomogenization for operators of Kolmogorov type. Our proof of the structuraltheorem is based on recent results established by the authors concerningboundary Harnack inequalities for operators of Kolmogorov type in divergenceform with bounded, measurable and uniformly elliptic coefficients.

A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type