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In this paper, we show that the so-called Korn inequality holds for vector fields with a zero normal or tangential trace on a subset (of positive measure) of the boundary of Lipschitz domains. We further show that the validity of this inequality depends on the geometry of this subset of the boundary. We then consider three eigenvalue problems for the Lamé operator: we constrain the traction in the tangential direction and the normal component of the displacement, the related problem of constraining the normal component of the traction and the tangential component of the displacement, and a third eigenproblem that considers mixed boundary conditions. We show that eigenpairs for these eigenproblems exist on a broad variety of domains. Analytic solutions for some of these eigenproblems are given on simple domains. [ABSTRACT FROM AUTHOR]

This paper introduces a simple variant of the power method. It is shown analytically and numerically to accelerate convergence to the dominant eigenvalue/eigenvector pair, and it is particularly effective for problems featuring a small spectral gap. The introduced method is a one-step extrapolation technique that uses a linear combination of current and previous update steps to form a better approximation of the dominant eigenvector. The provided analysis shows the method converges exponentially with respect to the ratio between the two largest eigenvalues, which is also approximated during the process. An augmented technique is also introduced and is shown to stabilize the early stages of the iteration. Numerical examples are provided to illustrate the theory and demonstrate the methods. [ABSTRACT FROM AUTHOR]

We present a novel computational framework for shape optimization problems arising in spectral geometry. The goal in such problems is to identify domains in R d which are the global optima of certain functions of the spectrum of elliptic operators on the domains. We propose the use of a combined finite element and Bayesian optimization (FEM–BO) framework in this context, and demonstrate the key ideas on two concrete examples. We study the Pólya–Szegö conjecture on polygons, and demonstrate that our proposed framework yields the theoretically proven result for triangles and quadrilaterals, and also provides compelling numerical evidence for the case of pentagons. We next study a variant of this conjecture for the Steklov eigenvalue problem. [ABSTRACT FROM AUTHOR]

Sharp upper bounds for the first eigenvalue of the Laplacian on a surface of a fixed area are known only in genera zero and one. We investigate the genus two case, and conjecture that the first eigenvalue is maximized on a singular surface which is realized as a double branched covering over a sphere. The six ramification points are chosen in such a way that this surface has a complex structure of the Bolza surface. We prove that our conjecture follows from a lower bound on the first eigenvalue of a certain mixed Dirichlet-Neumann boundary value problem on a half-disk. The latter can be studied numerically, and we present conclusive evidence supporting the conjecture. [ABSTRACT FROM AUTHOR]