Existence and structure of minimizers of least gradient problems
Existence and structure of minimizers of least gradient problems
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DOI：
10.1512/iumj.2018.67.7360
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发表时间：
2018-01-01
影响因子：
1.1
通讯作者：
Moradifam, Amir
中科院分区：
文献类型：
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作者：
Moradifam, Amir
本文研究了一般最小梯度问题minf（u是BV_f的一个元素）integral（Omega）phi（x，Du）的极小解的存在性，其中BVf = {u是BV（Omega）的一个元素：u|（偏导数Ω）= f}，f是L-1（偏导数Ω）的元素，并且phi（x，xi）是关于xi变量的1次凸的、连续的且齐次的函数。证明了存在一个无发散向量场T是（L-无穷（Ω））（n）的元素，它决定了所有（可能）极小元的水平集的结构，即T决定了Du/|杜|, |杜|- 几乎无处不在欧米茄，为所有的极简主义者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.inf的极小元（u是Af的元素）integral（n）（R）phi（x，Du），其中A（f）= {v是BV（R-n）的元素：v = f on Omega（c）}且f是W-1中的元素，W-1（R-n）是f是L-1中的元素的紧支撑扩张（偏导数Ω），并表明，T也决定了后一个问题的所有极小的水平集的结构。上述两个最小梯度问题的极小值之间的这种关系可以用来从后者的极小值的存在性和结构中获得关于前者的极小值的存在性和结构的信息，后者总是存在的。
We study existence of minimizers of the general least gradient probleminf(u is an element of BVf) integral(Omega) phi(x, Du),where BVf = {u is an element of BV(Omega) : u|(partial derivative Omega) = f}, f is an element of L-1(partial derivative Omega), and phi(x, xi) is a convex, continuous, and homogeneous function of degree 1 with respect to the xi variable. It is proven that there exists a divergence-free vector field T is an element of(L-infinity(Omega))(n) that determines the structure of level sets of all (possible) minimizers; that is, T determines Du/|Du|, |Du|-almost everywhere in Omega, for all minimizers u. We also prove that every minimizer of the above least gradient problem is also a minimizer of inf u.inf(u is an element of Af) integral(n)(R) phi(x, Du),where A(f) = {v is an element of BV(R-n) : v = f on Omega(c)} and f is an element of W-1,W-1(R-n) is a compactly supported extension of f is an element of L-1(partial derivative Omega), 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 those of the latter, which always exist.