Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems
Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems
复制标题
与最佳 Sobolev 迹常数相关的 Steklov 型特征值：域扰动和超定系统
DOI：
10.1080/17476933.2011.557155
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发表时间：
2011
影响因子：
0.9
通讯作者：
P. D. Lamberti
中科院分区：
文献类型：
--
作者：
P. D. Lamberti
我们考虑经典的Steklov特征值问题的一个变体，它出现在Sobolev空间中的函数的最佳迹常数的研究中。我们证明了初等对称函数的特征值依赖于实解析的基础域的变化，我们计算相应的Hadamard型公式的形状衍生物。我们还考虑等容和等周域扰动，我们相应的临界域适当的超定系统的特征。最后，我们证明了球是特征值的初等对称函数受体积或周长约束的临界域。
We consider a variant of the classic Steklov eigenvalue problem, which arises in the study of the best trace constant for functions in Sobolev space. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically upon variation of the underlying domain and we compute the corresponding Hadamard-type formulas for the shape derivatives. We also consider isovolumetric and isoperimetric domain perturbations and we characterize the corresponding critical domains in terms of appropriate overdetermined systems. Finally, we prove that balls are critical domains for the elementary symmetric functions of the eigenvalues subject to volume or perimeter constraint.