arXiv 1612.084002016-12-26math.AP

最小梯度问题极小解的存在性和结构

Existence and structure of minimizers of least gradient problems

Amir Moradifam

备注：: to appear in Indiana University Math Journal

摘要：

研究了一般最小梯度问题\[\inf_{u \in BV_f} \int_Ωφ(x,Du),\]最小解的存在性，其中$BV_f=\{u \in BV(Ω): \ \ u|_{\partial Ω}=f\}$，$f\in L^{1}(\partial Ω)$，和$φ(x,ξ)$是凸的，连续的，关于$ξ$变量的齐次函数.证明了存在一个无发散向量场$T\in (L^{\infty}(Ω))^n$，它决定了所有（可能的）极小元的水平集的结构，即$T$决定了$\frac{Du}{|Du|}$，$|Du|-$ a.e.在$Ω$中，对于所有极小化器$u$。我们还证明了上述最小梯度问题的每一个极小元也是\[\inf_{u\in \mathcal{A}_f} \int_{\R^n}φ(x,Du),\]的极小元，其中$\mathcal{A}_f=\{v\in BV(\R^n): \ \ v=f \ \ \hbox{on}\ \ Ω^c\}$和$f\in W^{1,1}(\R^n)$是$f\in L^1(\partial Ω)$的紧支撑扩张，并证明了$T$也决定了后一个问题的所有极小元的水平集结构。利用上述两个最小梯度问题的极小值之间的这种关系，可以从总是存在的最小梯度问题的极小值中获得关于前一个问题的极小值的存在性和结构的信息。

英文摘要：

We study existence of minimizers of the general least gradient problem \[\inf_{u \in BV_f} \int_Ωφ(x,Du),\] where $BV_f=\{u \in BV(Ω): \ \ u|_{\partial Ω}=f\}$, $f\in L^{1}(\partial Ω)$, and $φ(x,ξ)$ is convex, continuous, and homogeneous function of degree $1$ with respect to the $ξ$ variable. It is proven that there exists a divergence free vector field $T\in (L^{\infty}(Ω))^n$ that determines the structure of level sets of all (possible) minimizers, i.e. $T$ determines $\frac{Du}{|Du|}$, $|Du|-$ a.e. in $Ω$, for all minimizers $u$. We also prove that every minimizer of the above least gradient problem is also a minimizer of \[\inf_{u\in \mathcal{A}_f} \int_{\R^n}φ(x,Du),\] where $\mathcal{A}_f=\{v\in BV(\R^n): \ \ v=f \ \ \hbox{on}\ \ Ω^c\}$ and $f\in W^{1,1}(\R^n)$ is a compactly supported extension of $f\in L^1(\partial Ω)$, and show that $T$ also determines the structure of level sets of all minimizers of the latter problem. This relationship between minimizers of the above two least gradient problems could be exploited to obtain information about existence and structure of minimizers of the former problem from that of the latter, which always exist.

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摘要：

英文摘要：

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