Bounds for the first eigenvalue of Fredholm and elliptic operators

Adaptive Eigenvalue Computation for Elliptic Operators

This article is concerned with recent developments of adaptive wavelet solvers for elliptic eigenvalue problems. We describe the underlying abstract iteration scheme of the preconditioned perturbed iteration. We apply the iteration to a simple model problem in order to identify the main ideas which a numerical realization of the abstract scheme is based upon. This indicates how these concepts c...

A FREDHOLM THEORY FOR A CLASS OF FIRST-ORDER ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS IN Rn

The objects of interest are linear first-order elliptic partial differential operators with domain //i(/?" ; C*) in L2(Rn;Ck), the first-order coefficients of which become constant and the zero-order coefficient of which vanishes outside a compact set in R". It is shown that operators of this type are "practically" Fredholm in the following way: Such an operator has a finite index which is inva...

Estimates for the First Nonzero Eigenvalue of Elliptic Operators in Divergence Form

We consider elliptic operators in divergence form L = div (Φ · grad ) either on a closed Riemannian manifold or in a domain with compact closure and piecewise smooth boundary M where Φ : M → End(TM) is a positive definite symmetric smooth section of the bundle of all endomorphisms of TM . We show that the first nonzero L-eigenvalues in the closed or Dirichlet eigenvalue problems can be bounded ...

Bounds for Elliptic Operators in Weighted Spaces

Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized principal eigenvalue that we introduce here al...