论文标题
$ W^{1,1} $解决方案的存在，用于一类变异问题，并在凸域上线性增长
Existence of $W^{1,1}$ solutions to a class of variational problems with linear growth on convex domains
论文作者
论文摘要
我们考虑一类凸积分函数，该功能由参数梯度中的线性增长项组成，以及涉及与基准的$ l^2 $距离的保真度项。众所周知，此类功能可以在$ bv $空间中获得其INVIMA。在集成域是凸的假设下，我们证明，如果基准为$ w^{1,1} $，则该功能具有$ w^{1,1} $的最小化器。实际上，最小化器继承了$ w^{1，p} $从[1， +\ infty] $中的任何$ p \的基准中的规律性。在基准为$ bv $的情况下，我们还可以在最小化器的梯度的单数部分上获得定量限制。我们推断出线性生长基础功能的梯度流的类似结果。我们承认任何线性生长的凸积。
We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$ space. Under the assumption that the domain of integration is convex, we prove that if the datum is in $W^{1,1}$, then the functional has a minimizer in $W^{1,1}$. In fact, the minimizer inherits $W^{1,p}$ regularity from the datum for any $p \in [1, +\infty]$. We also obtain a quantitative bound on the singular part of the gradient of the minimizer in the case that the datum is in $BV$. We infer analogous results for the gradient flow of the underlying functional of linear growth. We admit any convex integrand of linear growth.
