|
A vast range of problems in applied mathematics can be modelled usingdifferential equations. Whilst simple models can be constructed using socalled `linear' differential equations, more realistic models usuallydemand the use of nonlinear equations. However, the nonlinearities insuch equations can range from being relatively simple to beinghopelessly complicated, so to make progress with any sort of theory ofsuch equations one must impose restrictions on the type of nonlinearityconsidered.A type of problem which is amenable to considerable analysis is one inwhich the nonlinearity is `asymptotically linear' - this means that whenthe unknown variable becomes large the problem essentially behaveslinearly, and we can then use the well understood linear theory toprovide information about the nonlinear problem. However, the assumptionof asymptotic linearity means that the nonlinear term behaves in thesame way when the unknown variable, say u, is both large and positive,and large and negative. Often, this is unrealistic, and in fact manyproblems have a certain type of (nearly) linear behaviour when u islarge and positive, and a different type of linear behaviour when u islarge and negative. Linearities with this type of behaviour are termed`jumping'. Jumping nonlinearities arise naturally in many applications,such as elasticity problems where the elastic constant involved isdifferent when the material is being stretched and compressed. Forexample, wires may be strong under tension, but have no resistance tocompression. Indeed, the theory of jumping nonlinearities has beenapplied to suspension bridges (a jumping nonlinearity model is relevanthere due to the cable supports for the road deck of the bridge), andleads to an explanation for the well-known Tacoma bridge collapse thatis different to the standardAlthough at first sight, jumping nonlinearities may seem to be only asimple generalisation of asymptotically linear problems, they are,surprisingly, much more complicated to deal with, and have been activelystudied since the late 70's. A generalisation of the usual idea of thespectrum of a linear operator to deal with jumping nonlinearities wasintroduced in about 1977. This was called the Fucik spectrum, and anextension of this, called the set of `half-eigenvalues' has been studiedmore recently. Each of these sets has been studied extensively, and leadto results that have no counterpart in linear or asymptotically linearproblems. In particular, for periodic problems the structure of theFucik spectrum or the set of half-eigenvalues is still not understood,although recent results show that this structure is much morecomplicated than in the linear case.Most work on these sets has dealt with the semilinear case, where astandard linear, second (or higher) order, elliptic differentialoperator has a jumping nonlinearity added to it. However, there is aquasilinear generalisation of such linear differential operators calledthe p-Laplacian. The p-Laplacian arises in many applications, such asnon-Newtonian fluid flows and percolation problems, and is currentlyunder intense study by mathematicians worldwide (as is shown bysearching for `p-Laplacian' on mathscinet). Many properties of the usualLaplacian extend to the p-Laplacian, but not all do so. In particular,since the p-Laplacian has a positive homogeneity property, it makessense to define a spectrum for it, and some of the spectral propertiesof the linear problem generalise to the p-Laplacian.It is our view that the interaction of the p-Laplacian and jumpingnon-linearities will provide fascinating and rich solution behaviour andwill provide a fundamental framework from which a better understandingof more complex applications can be obtained.
|