### A counterexample to the “hot spots" conjecture.

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This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound...

We control the gap between the mean value of a function on a submanifold (or a point), and its mean value on any tube around the submanifold (in fact, we give the exact value of the second derivative of the gap). We apply this formula to obtain comparison theorems between eigenvalues of the Laplace-Beltrami operator, and then to compute the first three terms of the asymptotic time-expansion of a heat diffusion process on convex polyhedrons in euclidean spaces of arbitrary dimension. We also write...

If the so-called Collatz method is applied to get twosided estimates of the first eigenvalue ${\lambda}_{1}$, the sequences of the so-called Schwarz quatients (which are upper bounds for ${\lambda}_{1}$) and of the so-called Temple quotients (which are lower bounds) are constructed. While monotony of the first sequence was proved many years ago, monotony of the second one has been proved only recently by F. goerisch and J. Albrecht in their common paper “Die Monotonie der Templeschen Quotienten” (ZAMM, in print). In the present...

We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on both Sasakian and 3-dimensional manifolds, and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.

We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators ${A}_{n}$, $n\in \mathbb{N}$, in a suitable Hilbert space. We show that the essential spectrum of ${A}_{n}$ is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators An, $n\in \mathbb{N}$, in a suitable Hilbert space. We show that the essential spectrum of An is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

In this paper, we present a superconvergence result for the mixed finite element approximations of general second order elliptic eigenvalue problems. It is known that a superconvergence result has been given by Durán et al. [Math. Models Methods Appl. Sci. 9 (1999) 1165–1178] and Gardini [ESAIM: M2AN 43 (2009) 853–865] for the lowest order Raviart-Thomas approximation of Laplace eigenvalue problems. In this work, we introduce a new way to derive the superconvergence of general second order elliptic...