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Nonlinear Boundary Value Problems
Nonlinear boundary value problems for both ordinary and partial differential
have been used to model an enormous variety of physical problems.
Questions of existence and uniqueness or multiplicity of solutions
of such problems are areas of major interest in current research
in `applied analysis'.
The following are three areas in the general field of nonlinear boundary
value problems in which there are many interesting problems.
Research in these areas would require a general knowledge of differential
equations and also a certain amount of functional analysis and topology.
Bifurcation Theory
A lot is known about local and global bifurcation theory of nonlinear
boundary value problems for both ordinary and partial differential
equations on bounded regions.
Less is known about corresponding problems on unbounded regions, and
there are many open questions in this area.
Generic results
Another fruitful approach to nonlinear problems is, instead of looking for
theorems that apply to `all' problems, to look for results that hold for a
`generic' or `typical' class of problems.
Such results are often easier to obtain than more general results, and also,
hopefully, say more about what should actually happen `in practice'.
Jumping nonlinearities
Nonlinearities which behave differently (asymmetrically)
depending on whether the solution is positive or negative are termed
`jumping nonlinearities'.
Such asymmetric nonlinearities often arise in elasticity when materials
have different elastic constants under extension and compression.
For instance, in an extreme form, when considering a particle attached
to a thin wire which has a strong restoring force under extension,
but no restoring force under compression
(such equations have been used in modelling suspension bridges where
such asymmetries between extension and compression arise when modelling
the oscillation of the road deck suspended from cables).
If a jumping nonlinearity is `asymptotically homogeneous' then a
generalised spectrum can be defined for the problem, and this can then
be used to investigate existence and multiplicity of solutions.
However, for all but the simplest problems, the full structure of this spectrum is not known, but is under active investigation at present.