发表机构
The University of Edinburgh; Maxwell Institute of Mathematical Sciences（爱丁堡大学; 麦克斯韦数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决时变图域上抛物型正则性边界空间定义依赖图参数化的问题，证明Lewis--Murray条件保证一致性，并将Dirichlet-正则性对偶性推广到此类域。
AI 中文摘要
本文研究了一个在抛物型正则性问题文献中尚未被考虑的基本问题。设$\Omega$为$\mathbb R^{n-1}\times\mathbb R$上$\mathrm{Lip}(1,\tfrac12)$函数$\phi$的图上方区域。相应的$\partial\Omega$上的$\dot L^p_{1,1/2}$正则性边界空间通过将数据拉回到平坦空间来定义，在平坦空间中范数为$\\|\nabla f\\|_{L^p}+\\|D^{1/2}_t f\\|_{L^p}$。由于此类域存在不止一种图参数化，这可能导致$\dot L^p_{1,1/2}(\partial\Omega)$的不同定义。该问题仅出现在时变域上，这很可能是之前未被注意的原因。我们通过例子表明边界空间确实可能不一致。然后我们证明附加假设$D^{1/2}_t\phi\in\mathrm{BMOpar}$（称为Lewis--Murray条件）完全解决了该问题：通过不同图定义的空间是一致的。这给出了在每个Lewis--Murray图域上良定义的正则性数据空间。接着我们将作者与E.~Nyström在柱体上证明的抛物型Dirichlet问题与正则性问题之间的对偶性推广到此类域。
英文摘要
In this paper we address a basic question that has not been considered in the literature on the parabolic Regularity problem. Let $Ω$ be the region above the graph of a $\mathrm{Lip}(1,\tfrac12)$ function $ϕ$ on $\mathbb R^{n-1}\times\mathbb R$. The corresponding $\dot L^p_{1,1/2}$ Regularity boundary space on $\partialΩ$ is defined by pulling the datum back to the flat space where we have the norm $\|\nabla f\|_{L^p}+\|D^{1/2}_t f\|_{L^p}$. As there is more than one graph parametrisation of such a domain, this could petentially lead to different notions of $\dot L^p_{1,1/2}(\partialΩ)$. This issue arises only on time-varying domains, which is likely why the issue has not been noticed before. We show by example that the boundary spaces can indeed disagree. We then prove that an additional assumption $D^{1/2}_tϕ\in\mathrm{BMOpar}$ (called the Lewis--Murray condition), resolves the issue completely: the spaces defined through different graphs agree.\medskip This gives us a well-defined Regularity datum space on every Lewis--Murray graph domain. We then extend the duality between the parabolic Dirichlet and Regularity problems, proved on cylinders by the author and E.~Nyström, to such domains.
Comments39 pages
