SPECTRAL STABILITY ESTIMATES FOR ELLIPTIC OPERATORS SUBJECT TO DOMAIN TRANSFORMATIONS WITH NON‐UNIFORMLY BOUNDED GRADIENTS

Multilevel Methods for Nonuniformly Elliptic Operators

We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an A2–Muckenhoupt weight. Using the so-called Xu–Zikatanov (XZ) identity, we derive a nearly uniform convergence result, under the assumption that the underlying mesh is quasi-uniform. We also consider the so-called α-harmonic extension to localize fractional po...

V. I. Burenkov RECENT PROGRESS IN THE PROBLEM OF SPECTRAL STABILITY ESTIMATES FOR ELLIPTIC OPERATORS

Lp Spectral Independence of Elliptic Operators via Commutator Estimates

Let fT p : q 1 p q 2 g be a family of consistent C 0 semigroups on L p ((), with q 1 ; q 2 2 1; 1) and R n open. We show that certain commutator conditions on T p and on the resolvent of its generator A p ensure the p independence of the spectrum of A p for p 2 q 1 ; q 2 ]. Applications include the case of Petrovskij correct systems with HH older continuous coeecients, Schrr odinger operators, ...

Multilevel methods for nonuniformly elliptic operators and fractional diffusion

We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an A2–Muckenhoupt weight. Using the so-called Xu-Zikatanov (XZ) identity, we derive a nearly uniform convergence result under the assumption that the underlying mesh is quasi-uniform. As an application we also consider the socalled α-harmonic extension to locali...

The spectral properties of differential operators with matrix coefficients on elliptic systems with boundary conditions

Let $$(Lv)(t)=sum^{n} _{i,j=1} (-1)^{j} d_{j} left( s^{2alpha}(t) b_{ij}(t) mu(t) d_{i}v(t)right),$$ be a non-selfadjoint differential operator on the Hilbert space $L_{2}(Omega)$ with Dirichlet-type boundary conditions. In continuing of papers [10-12], let the conditions made on the operator $ L$ be sufficiently more general than [11] and [12] as defined in Section $1$. In this paper, we estim...