On the effect of inhomogeneous constraints for a maximizing problem associated with the Sobolev embedding of the space of functions of bounded variation
On the effect of inhomogeneous constraints for a maximizing problem associated with the Sobolev embedding of the space of functions of bounded variation
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DOI：
10.4064/sm190613-13-7
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发表时间：
2021
影响因子：
0.8
通讯作者：
M. Ishiwata;H. Wadade
中科院分区：
文献类型：
--
作者：
M. Ishiwata;H. Wadade
.我们考虑一个与Sobolev型嵌入BV（R N）（cid：44）→ L r（R N）相关的最大化问题，其中1 ≤ r ≤ 1：= NN − 1，N ≥ 2。对于给定的α > 0，设置D α（a，B，q）：我们证明了，虽然D α（a，B，1 λ）的极大化问题是BV（cid：44）→ L1和BV（cid：44）→ L1 λ的非紧性的一个子问题，称为消失和集中现象，但在a，B的一定范围内，存在一个极大化子.进一步证明了D α（a，B，q）的任意极大元u ∈ BV都由球的特征函数给出.
. We consider a maximizing problem associated with the Sobolev type embedding BV( R N ) (cid:44) → L r ( R N ) for 1 ≤ r ≤ 1 ∗ := NN − 1 with N ≥ 2 . For given α > 0 , set D α ( a, b, q ) := sup where 1 0 . We show that, although the maximizing problem associated with D α ( a, b, 1 ∗ ) suffers from both of the non-compactness of BV (cid:44) → L 1 and BV (cid:44) → L 1 ∗ , called the vanishing and concentrating phenomena, there exists a maximizer for some range of a , b . Furthermore, we show that any maximizer u ∈ BV of D α ( a, b, q ) is given by the characteristic function of a ball.