On the placement of an obstacle so as to optimize the Dirichlet heat trace (Q2793833)

For a bounded \(C^2\) domain \(\Omega\) in \(\mathbb R^n\), let \(\lambda_k (\Omega)\) be the increasing sequence (repeated according to its multiplicity) of Dirichlet eigenvalues of the Laplacian \(-\Delta\) in \(\Omega\). The trace of the heat operator is: \(Z_\Omega (t)=\sum_{k\geq 1} e^{-\lambda_k(\Omega)t}\); the spectral zeta function is: \(\zeta_\Omega (s)=\sum_{k=1}^\infty \lambda_k(\Omega)^{-s}\); the regularized determinant is: \(\det(\Omega)=\exp(- \zeta_\Omega'(0))\). Given \(R>r>0\), \(x\in\mathbb R^n\) with \(|x|<R-r\), denote \(\Omega(x)\) be the domain of \(\mathbb R^n\) obtained by removing the ball \(B(x, r)\) of radius \(r\) centered at \(x\) from within the ball of radius \(R\) centered at the origion. Mainly, in this paper, some Hersch-type extremal results for the heat trace, the spectral zeta function, and the determinant of the Laplacian of \(\Omega(x)\) are given. Among \(\Omega(x)\), the authors proved that \(Z_{\Omega(x)}(t)\), \(\forall t>0\), is nondecreasing; \(\zeta_{\Omega(x)}(s)\), \(\forall s>0\), increases; and det\(( \Omega(x))\) decreases respectively as the point \(x\) moves from the origin directly toward the boundary of the larger ball. Similar results are given with suitable generalization for more general outer domains.