In this article, we study the ergodic risk-sensitive control problem for controlled regime-switching diffusions. Under a blanket stability hypothesis, solve associated nonlinear eigenvalue weakly coupled systems and characterize optimal stationary Markov controls via suitable verification theorem. We also consider near-monotone case obtain existence of principal eigenfunction controls.

In this paper we propose and analyze a hp-adaptive discontinuous finite element method for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectrum of each member in a family of periodic Hermitian eigenvalue problems on the primitive cell, parametrised by a two-dimensional parameter the quasimomentum. We propose a resi...

Abstract — In this article, we address the numerical solution of a non-smooth eigenvalue problem, which has implications in plasticity theory and image processing. The smallest eigenvalue of the nonsmooth operator under consideration is shown to be the same for all bounded, sufficiently smooth, domains in two space dimensions. Piecewise linear finite elements are used for the discretization of ...

This paper introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded m...

Abstract We study the eigenvalues and eigenfunctions of one-dimensional weighted fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps. first prove existence eigenfunctions. then set up a framework for to discretize equation defining eigenfunctions, obtain numerical approximations eigenvalue eigenfunction using finite element method. Finally, we show that conve...

Let Ω be a planar domain such that Ω is symmetric with respect to both the xand y-axes and Ω satisfies certain conditions. Then the second eigenvalue of the Dirichlet Laplacian on Ω, ν2(Ω), is simple, and the corresponding eigenfunction is odd with respect to the y-axis. Let f ∈ C3 be a function such that f ′(0) > 0, f ′′′(0) < 0, f(−u) = −f(u) and d du ( f(u) u ) < 0 for u > 0. Let C denote th...

Abstract A numerical scheme for solving high-dimensional stochastic control problems on an infinite time horizon that appear relevant in the context of molecular dynamics is outlined. The scheme rests on the interpretation of the corresponding Hamilton-Jacobi-Bellman equation as a nonlinear eigenvalue problem that, using a logarithmic transformation, can be recast as a linear eigenvalue problem...