Abstract In this paper, we consider the first eigenvalue λ 1 ⁢ ( Ω stretchy="false">) {\lambda_{1}(\Omega)} of Grushin operator mathvariant="normal">Δ G :=</m:...

Michel Baes (ETH Zurich) First-order methods for eigenvalue minimization Many semidefinite programming problems encountered in practice can be recast as minimizing the maximal eigenvalue of a convex combination of symmetric matrices. In this talk, we describe and analyze a series of first-order methods for solving this problem when the input matrices are large (of dimension 1000 to 10000 and mo...

ABSTRACT : The stability operator of a compact oriented minimal hypersurface Mn−1 ⊂ S is given by J = −∆ − ‖A‖ − (n − 1), where ‖A‖ is the norm of the second fundamental form. Let λ1 be the first eigenvalue of J and define β = −λ1 − 2(n − 1). In [S] Simons proved that β ≥ 0 for any non-equatorial minimal hypersurface M ⊂ S. In this paper we will show that β = 0 only for Clifford hypersurfaces. ...

For $N\geq 2$ , a bounded smooth domain $\Omega$ in $\mathbb {R}^{N}$ and $g_0,\, V_0 \in L^{1}_{loc}(\Omega )$ we study the optimization of first eigenvalue for following weighted problem: \[ -\Delta_p \phi + V |\phi|^{p-2}\phi = \lambda g \text{ } \Omega, \quad \phi=0 on \partial \] where $g$ $V$ vary over rearrangement classes $g_0$ $V_0$ respectively. We prove existence minimizing pair $(\u...

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

In this short note we extend results of Kohnen and Sengupta on the sign of eigenvalues of Siegel cuspforms. We show that their bound for the first negative Hecke eigenvalue of a genus 2 Siegel cuspform of level 1 extends to the case of level N > 1. We also discuss the signs of Hecke eigenvalues of CAP forms.