论文标题
$ \ mathbb {r}^n $中的四阶总变化流
The fourth-order total variation flow in $\mathbb{R}^n$
论文作者
论文摘要
我们严格地定义了$ \ mathbb {r}^n $中四阶总变化流程方程的解决方案。如果$ n \ geq3 $，则可以理解为$ d^{ - 1} $中总变化能量的梯度流，这是$ d^1_0 $的双空间，这是在dirichlet norm中的紧凑型平滑功能的完成。但是，在低维情况$ n \ leq2 $中，空间$ d^{ - 1} $不包含积极度量集的特征功能，因此我们将解决方案的概念扩展到了更大的空间。我们根据双重性参数来表征解决方案的解决方案。该论点依赖于近似引理本身很有趣。 我们在我们的四阶设置中介绍了集合的校准性概念。该概念与特征函数是否在整个进化过程中保存形式有关。事实证明，所有球都是可校准的。但是，与二阶总变化流不同，只有$ n \ neq2 $，球的外部才能校准。如果$ n \ neq2 $，则所有Annuli都是可校准的，而在情况下，如果环形太厚，则无法校准。 我们从球的特征函数中明确计算溶液。我们还提供了从任何分段常数，径向对称基准中发出的解决方案的描述。
We define rigorously a solution to the fourth-order total variation flow equation in $\mathbb{R}^n$. If $n\geq3$, it can be understood as a gradient flow of the total variation energy in $D^{-1}$, the dual space of $D^1_0$, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $n\leq2$, the space $D^{-1}$ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $n\neq2$. If $n\neq2$, all annuli are calibrable, while in the case $n=2$, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.
