Unilateral global bifurcation and nodal solutions for the pp-Laplacian with sign-changing weight

In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(\mu_k^\nu(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^\nu)^+$ and $(\mathcal{C}_{k}^\nu)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^\nu$ from $(\mu_k^\nu(p), 0)$, where $\mu_k^\nu(p)$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $\nu\in\{+,-\}$. As the applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Dr\'{a}bek and Huang (1997) [\ref{DH}], we study the existence of one-sign solutions for a class of high dimensional quasilinear elliptic problems with sign-changing weight

Quasilinear eigenvalues

In this work, we review and extend some well known results for the eigenvalues of the Dirichlet $p-$Laplace operator to a more general class of monotone quasilinear elliptic operators. As an application we obtain some homogenization results for nonlinear eigenvalues.Comment: 23 pages, Rev. UMA, to appea

A new level-dependent coarsegrid correction scheme for indefinite Helmholtz problems

In this paper we construct and analyse a level-dependent coarsegrid correction scheme for indefinite Helmholtz problems. This adapted multigrid method is capable of solving the Helmholtz equation on the finest grid using a series of multigrid cycles with a grid-dependent complex shift, leading to a stable correction scheme on all levels. It is rigourously shown that the adaptation of the complex shift throughout the multigrid cycle maintains the functionality of the two-grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state-of-the-art multigrid-preconditioned Krylov methods, like e.g. CSL-preconditioned GMRES or BiCGStab.Comment: 21 page

Precise homogenization rates for the Fučík spectrum

Given a bounded domain Ω in RN, N≥ 1 we study the homogenization of the weighted Fučík spectrum with Dirichlet boundary conditions. In the case of periodic weight functions, precise asymptotic rates of the curves are obtained.Fil: Salort, Ariel Martin. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; Argentin

Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page