Training content (objective, benefit and expected impact) This research project requires a considerable amount of analysis, especially

elliptic operator theory.

In Section 3 we extend the obtained results to the case of the nonlinear

elliptic operator Au [equivalent to] -[nabla] (k([[absolute value of [nabla]u].

For instance, such a structure arises when decomposing the domain of definition of an

elliptic operator using unidirectional stripes, or more generally, for a decomposition such that (in addition to a corresponding portion of the original boundary) each subdomain has a common boundary only with its previous and next neighbours in the sequence of subdomains.

i] of the linear

elliptic operator are piecewise constant and present jump discontinuities across subdomain boundaries.

Among the topics are variation on the p-Laplacian, extremal functions in Poincare-Sobolev inequalities for functions of bounded variation, homocline type solutions for a class of differential equations with periodic coefficients, the cooperative case of quasilinear and singular systems, weighted asymmetric problems for an indefinite

elliptic operator, multiple non-trivial solutions of the Dirichlet problem for the prescribed mean curvature equation, and existence of nodal solutions for some nonlinear elliptic problems.

Bounds on eigenvalues which are independent of both degrees of high-order elements and mesh sizes are shown for the system preconditioned by bilinear elements for high-order finite element discretizations applied to a model uniformly

elliptic operator.

STRAKHOVSKAYA, An iterative method for evaluating the first eigenvalue of an

elliptic operator, USSR Comput.

Peletier: An anti-maximum principle for second-order

elliptic operators, J.

Among them are entanglement protection and generation under continuous monitoring, completely positive transformations of quantum operations, generating semigroups by degenerate

elliptic operators arising in open quantum systems, the spectral gap of the n-photon absorption-emission process, white noise theory, computational complexity of a quantum algorithm for factoring, an isometry formula for a new stochastic integral, and complexities for Gaussian communication processes.

Elliptic operators and boundary value problems were studied by [2, 10, 12], hypoelliptic and hyperbolic equations with constant coefficients were developed by Hormander [8].

They cover Hardy and Hardy-Rellich types of inequalities, Hardy inequalities for general

elliptic operators, Hardy-Rellich-Sobolev inqualities, and Aubin-Moser-Onofri inequalities.

A sampling of topics includes: the analog of the limiting absorption principle in homeogenization of periodic

elliptic operators, spectral instability of semiclassical operators, Green's kernels for transmission problems in bodies with small incisions, and some open problems in spectral theory.