Charles Alexander Stuart

Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity

In this paper, we prove the uniqueness of the solution to certain simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for a body occupying a star-shaped reference configuration Ω ⊂ Rn whose boundary ∂ Ω is subjected to an affine deformation i.e., there exists a constant n×n matrix F and a constant n vector b such that x ↦ Fx + b for all x∈ ∂ Ω. We consider all smooth equilibrium configurations satisfying this boundary condition. Clearly, the homogeneous deformation x ↦ Fx + b, for all \( x \in \bar{\Omega } \), is one such solution. Our aim is to prove under suitable hypotheses that it is the only such solution.

Radially symmetric cavitation for hyperelastic materials

Keywords: radially symmetric cavitation ; hyperelastic material ; nonlinear boundary value problem ; singular second order ; differential equation ; class of stored-energy ; densities ; existence ; class of singular radial ; solutions ; equilibrium equations ; spherical hole in a ; ball ; isotropic material ; shooting method Reference ANA-ARTICLE-1985-002View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

Guidance properties of nonlinear planar waveguides

AbstractWe discuss the propagation of electromagnetic waves through a stratified dielectric. The ability of such a device to support guided waves depends upon the way in which the refractive index varies across the layers. In the present discussion, we show how nonlinear effects and appropriate stratification can be used to obtain any one of the following behaviours:(i)guidance occurs only at low power.(ii)guidance occurs only at high power.(iii)guidance occurs at all powers.(iv)there is no guidance. The situation (i) is obtained by using materials with a defocusing dielectric response, whereas the situation (ii) is obtained for suitable configurations of self-focusing materials. The situations (iii) and (iv) can be obtained by using either defocusing or self-focusing materials.By seeking solutions of a particular form, we reduce the problem to the study of solutions in the Sobolev space H1(ℝ) of a second-order differential equation.The discussion of defocusing nonlinearities is based in the study of the global behaviour of the branch of solutions bifurcating from a simple eigenvalue. For self-focusing nonlinearities we use a variational approach.

Self-trapping of an electromagnetic field and bifurcation from the essential spectrum

Keywords: bifurcation from the essential spectrum ; exact solutions ; of Maxwells equations Reference ANA-ARTICLE-1991-001doi:10.1007/BF00380816View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

A Variational Problem Related to Self-trapping of an Electromagnetic Field

Reference ANA-ARTICLE-2009-002doi:10.1002/(SICI)1099-1476(19961125)19:17 3.0.CO;2-B Record created on 2009-02-17, modified on 2016-08-08

Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation

Abstract In this paper we apply previously obtained abstract bifurcation results to nonlinear perturbations of the periodic Schrodinger equation. We show that (depending on the sign of the nonlinearity) lower or upper end-points of the continuous spectrum are bifurcation points. The main part of the proof consists of the construction of suitable test-functions which are required for the application of the abstract theorem. Instead of pursuing the notion of an eigenpaket of the differential operator which has successfully been used in the one-dimensional case we now exploit a construction based on Bloch waves of the linear Schrodinger operator.

Fredholm and Propernes Properties of Quasilinear Elliptic Operators on RN

Keywords: topological degree techniques ; Frechet derivative ; asymptotic behavior of coefficients Reference ANA-ARTICLE-2001-002doi:10.1002/1522-2616(200111)231:1 3.0.CO;2-VView record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

Bifurcation for variational problems when the linearisation has no eigenvalues

Keywords: bifurcation ; non-linear Klein-Gordon equation Reference ANA-ARTICLE-1980-002doi:10.1016/0022-1236(80)90063-4View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

Global bifurcation for quasilinear elliptic equations on

Abstract. In this paper we discuss the global behaviour of some connected sets of solutions

R^N

(\lambda,u)