Laplace型算子的广义紧p-框架和谱界
Generalized tight p-frames and spectral bounds for Laplace-like operators
摘要：
我们证明了尖锐的上界的特征值（和其他谱泛函）的拉普拉斯算子，包括双拉普拉斯算子和分数拉普拉斯算子的总和。我们表明，在一个高度对称的域的线性图像，我们的谱泛函是最大的原始域。我们利用域的对称性和算子，避免了为变分问题寻找好的检验函数的必要性。这对于分数拉普拉斯算子尤其重要，因为在区间上甚至不知道精确解，使得很难找到好的检验函数。为了实现我们的目标，我们推广紧密的$p$融合框架，提取最好的几何结果域的等距组承认紧密的$p$框架。任何这样的群通过任意投影矩阵的共轭生成紧密的$p$ -融合框架。我们证明了广义紧$p$ -框架也可以通过任意矩形矩阵的共轭得到，框架常数依赖于矩阵的奇异值。
英文摘要：
We prove sharp upper bounds for sums of eigenvalues (and other spectral functionals) of Laplace-like operators, including bi-Laplacian and fractional Laplacian. We show that among linear images of a highly symmetric domain, our spectral functionals are maximal on the original domain. We exploit the symmetries of the domain, and the operator, avoiding necessity of finding good test functions for variational problems. This is especially important for fractional Laplacian, since exact solutions are not even known on intervals, making it hard to find good test functions. To achieve our goals we generalize tight $p$-fusion frames, to extract the best possible geometric results for domains with isometry groups admitting tight $p$-frames. Any such group generates a tight $p$-fusion frame via conjugation of arbitrary projection matrix. We show that generalized tight $p$-frames can also be obtained by conjugation of arbitrary rectangular matrix, with frame constant depending on the singular values of the matrix.
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