论文标题
抛物线最小化器的存在至公制空间的总变化流量
Existence of parabolic minimizers to the total variation flow on metric measure spaces
论文作者
论文摘要
我们为与总变异流相关的各种解决方案$ u $提供了证明。在这里，所考虑的功能是在公制度量空间$（\ Mathcal {x}，d，μ）$上定义的，可满足双倍条件并支持庞加莱的不平等。对于与时间无关的cauchy-dirichlet基准$ u_0 $相吻合的抛物面最小化器，在时空缸$ω\ times（0，t）$上与$ω\ subset \ subset \ subset \ mathcal {x} $ a Open nep and a Open集合和$ t $ t> 0 $ nek parababoric function y live parababioloical Function y light-subset \ subset \ subset \ subset \ mathcal {x} $ l^l. \ mathrm {bv}（ω））$。在本文中，我们通过引入$ \ mathrm {bv} $ - 有价值的抛物线函数空间的更抽象的概念来概括Bögelein，Duzaar和Marcellini先前的作品的结果。我们完全在各个层面上争论。
We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(\mathcal{X}, d, μ)$ satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $Ω\times (0, T)$ with $Ω\subset \mathcal{X}$ an open set and $T > 0$, we prove existence in the weak parabolic function space $L^1_w(0, T; \mathrm{BV}(Ω))$. In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $\mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.
