In the last decade, there has been an increasing interest in the p-Laplacian, which plays an important role in geometry and partial differential equations. The p-Laplacian is a natural generalization of the Laplacian. Although the Laplacian has been much studied, not much is known about the nonlinear case p >1. Motivated by these facts, the purpose of the present paper is to review recent developments in the spectral theory of a specific class of quantum waveguides modeled by the Dirichlet Laplacian, i.e. p = 2, in unbounded tubes of uniform cross-section rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension. We discuss how the spectrum depends upon three geometric deformations: straightness, asymptotic straightness, and bending. Precisely, if the reference curve is straight or asymptotic straight, the essential spectrum is preserved. While dealing with bent tubes, such geometry produces a spectrum below the first eigenvalue. All the results confirm the literature for the Laplacian operator. The results are obtained via a very delicate analysis since the nonlinearity given by the p-Laplacian operator adds different types of difficulties with respect to the linear situation. These results are contained in a work written jointly with D. Krejčiřík.[-]

Polyakov's formula expresses a difference of zeta-regularized determinants of Laplace operators, an anomaly of global quantities, in terms of simple local quantities. Such a formula is well known in the case of closed surfaces (Osgood, Philips, & Sarnak 1988) and surfaces with smooth boundary (Alvarez 1983). Due to the abstract nature of the definition of the zeta-regularized determinant of the Laplacian, it is typically impossible to compute an explicit formula. Nonetheless, Kokotov (genus one Kokotov & Klochko 2007, arbitrary genus Kokotov 2013) demonstrated such a formula for polyhedral surfaces ! I will discuss joint work with Clara Aldana concerning the zeta regularized determinant of the Laplacian on Euclidean domains with corners. We determine a Polyakov formula which expresses the dependence of the determinant on the opening angle at a corner. Our ultimate goal is to determine an explicit formula, in the spirit of Kokotov's results, for the determinant on polygonal domains.[-]