论文标题
公制度量空间中的抛物线差异问题的时间平滑
Time-smoothing for parabolic variational problems in metric measure spaces
论文作者
论文摘要
2013年，Masson和Siljander确定了一种方法来证明$ p $ - 最小的上部梯度$ g_ {f_ \ varepsilon} $ time Mollification $ f_ \ f_ \ varepsilon $，$ \ varepsilon> 0 l^p_ \ mathrm {loc}（0，τ; n^{1，p} _ \ mathrm {loc}（ω））$，$τ> 0 $和$ω$ open域中的$ tops $（\ mathbb {x}，d，d，d，d，d，d，μ $ p \ in（1，\ infty）$，以至于$ g_ {f-f_ \ varepsilon} \ to0 $ as $ \ varepsilon \ to in $ l^p_ \ p_ p_ \ mathrm {loc}（loc}（ω____________________________________________________________________________________________________________）他们的方法涉及使用Cheeger的差异结构，因此显示出一些局限性。在这里，我们将看到，抛物线索波列夫空间的定义和形式属性本身可以找到一种更直接的方法来显示这种收敛性，该方法仅依赖于$ p $ - 净值上梯度，而无论在$ p = 1 $时，在限制情况下，在环境空间上的结构假设与环境空间的结构假设无关。
In 2013, Masson and Siljander determined a method to prove that the $p$-minimal upper gradient $g_{f_\varepsilon}$ for the time mollification $f_\varepsilon$, $\varepsilon>0$, of a parabolic Newton-Sobolev function $f\in L^p_\mathrm{loc}(0,τ;N^{1,p}_\mathrm{loc}(Ω))$, with $τ>0$ and $Ω$ open domain in a doubling metric measure space $(\mathbb{X},d,μ)$ supporting a weak $(1,p)$-Poincaré inequality, $p\in(1,\infty)$, is such that $g_{f-f_\varepsilon}\to0$ as $\varepsilon\to0$ in $L^p_\mathrm{loc}(Ω_τ)$, $Ω_τ$ being the parabolic cylinder $Ω_τ:=Ω\times(0,τ)$. Their approach involved the use of Cheeger's differential structure, and therefore exhibited some limitations; here, we shall see that the definition and the formal properties of the parabolic Sobolev spaces themselves allow to find a more direct method to show such convergence, which relies on $p$-weak upper gradients only and which is valid regardless of structural assumptions on the ambient space, also in the limiting case when $p=1$.
