论文标题
Sobolev Spacesi的嵌入恒定问题中极端功能的明确形式
Explicit Form Of Extremal Functions In The Embedding Constant Problem For Sobolev SpacesI
论文作者
论文摘要
Sobolev Spaces的嵌入常数$ \ MATHRING {w}^n_2 [0; 1] \ hookrightArrow \ Mathring \ Mathring {w}^k_ \ infty [0; 1] $（$ 0 \ leqslant k \ leqslant n-1 $）。嵌入常数与功能的规范的关系$ f \ mapsto f^{（k）}（a）$在space $ \ mathring {w}^n_2 [0; 1] $中。函数的明确形式$ g_ {n; k} \ in \ mathring {w}^n_2 [0; 1] $，这些功能可以找到它们的标准。这些功能对于嵌入常数也是极端的。提出了嵌入常数与Legendre多项式的关系。一项详细的研究是针对具有k = 3和k = 5的嵌入常数进行的：我们发现了极端点的明确公式，计算全局最大点并给出尖锐嵌入常数的值。发现了嵌入常数与某些光谱问题与分布系数之间的联系。
The embedding constants of the Sobolev spaces $\mathring{W}^n_2[0;1] \hookrightarrow \mathring{W}^k_\infty[0; 1]$ ($0\leqslant k \leqslant n-1$) are studied. A relation of the embedding constants with the norms of the functionals $f\mapsto f^{(k)}(a)$ in the space $\mathring{W}^n_2[0;1]$ is given. An explicit form of the functions $g_{n;k}\in \mathring{W}^n_2[0;1]$ on which these functionals attain their norm is found. These functions are also to be extremal for the embedding constants. A relation of the embedding constants to the Legendre polynomials is put forward. A detailed study is made of the embedding constants with k = 3 and k = 5: we found explicit formulas for extreme points, calculate global maximum points, and give the values of the sharp embedding constants. A link between the embedding constants and some class of spectral problems with distribution coefficients is discovered.
