Research in Mathematical Analysis[Department of Mathematical Sciences] [University of Bath] [Postgraduate] [Research]Staff working in this area are Dr
G. R. Burton, Professor L. E. Fraenkel FRS,
Dr J. Sivaloganathan, Professor
J. F. Toland FRS and The topics studied include Convex
analysis and variational principles, Free-boundary
problems, Mathematical hydrodynamics,
Nash-Moser theory, Nonlinear equations involving the Hilbert transform. Research in analysis at Bath is mainly concerned with
the rigorous mathematical theory of partial differential equations
which arise in problems from mechanics, physics and geometry, with
special emphasis on nonlinear problems. Questions about differential equations can be formulated in
terms of operators acting on Banach spaces of functions. There are
natural connections with differential geometry, nonlinear functional analysis, complex analysis, dynamical systems
theory and harmonic analysis, which are reflected in the wide range of
work being done here, including collaborations with colleagues in other
groups. There is a lively research environment with regular seminars
and international visitors. Graduate lectures and research at Bath led
to the first book on a recent theory of symmetry of positive solutions of elliptic partial differential equations. Convex sets and convex functionals play a prominent role in
the modern theory of the Calculus of Variations. For example, the
special properties enjoyed by convex sets and functionals with respect to weak convergence in Banach spaces, that result from
the Hahn-Banach theorem, are often crucial in proving the existence of
minimizers for variational problems. The subject has advanced to the stage where many interesting non-convex optimization
problems can be studied using convexity methods. Examples are the modern
nonlinear models of elasticity using polyconvex and quasiconvex
functionals, and the variational formulations of steady fluid vortices where the (non-convex) set of rearrangements of a
fixed function arises as a constraint set. For other rigorous work on the calculus of variations.
See Nonlinear
Elasticity. The Hilbert transform is a singular integral operator
from harmonic analysis which acts on spaces of Lebesgue measurable
functions. In a very natural way it arises in a number of nonlinear equations, such as the Peierls-Nabarro equation
and Benjamin-Ono equation from mechanics. These equations involve self-adjoint operators that are not differential operators and
we are interested in developing techniques for understanding general
classes of such equations from the viewpoint of bifurcation theory, regularity theory, variational methods and degree theory. Linear operators which are invertible but have unbounded inverses in their natural setting arise in lots of
problems, including problems from dynamical systems theory. The usual
implicit function theorem is then no longer applicable but in important cases it can be replaced with a much more delicate
and technically demanding tool, known under the general title of
Nash-Moser theory. Recently, workers at Bath in collaboration with
colleagues in Russia have used this method to give the first proof of the existence of standing waves on deep water. A great deal of analysis research at Bath is motivated
by problems from mechanics, and especially fluid mechanics. Of
particular is the mathematics involved in the theory of hydrodynamic waves and other free-boundary problems, and
vortex flow problems. Whitney showed that any closed set in the plane is
the solution set of the equation f(x) = 0 for some infinitely
differentiable function f. By contrast, the complement of the zero set of a real-analytic function f is dense unless
f is identically zero. When f maps between Euclidean
spaces and is real analytic, its zero set is a real-analytic
variety, and the structure of such sets is well understood. Here we are interested in the implications of these structure
theorems when f is a real-analytic function between
infinite-dimensional Banach spaces. A great deal more can be said
globally about the solution set of f(x) = 0 in this case. [B. Buffoni and J. F. Toland: Global real-analytic
bifurcation theory - an introduction (Princeton University Press,2002).] The classical Riemann-Hilbert problem is to find all pairs
of bounded holomorphic functions, one on the interior and one on the
exterior of the unit disc in the complex plane such that their ratio on the unit circle is a prescribed positive real
function. However certain nonlinear free boundary problems for solutions
of Laplace's equation give rise naturally to Riemann-Hilbert problems
where the ratio function is not strictly positive, and may change sign. We have been developing a general theory for
such problems. The celebrate work of Gidas, Ni and Nirenberg on
positive solutions of nonlinear elliptic equations has been extended
from classical to generalized solutions (in order to make it applicable
to certain physical problems) and elementary proofs have been found for some basic theorems of this subject. [L. E. Fraenkel:
An introduction to maximum principles and symmetry in elliptic
problems (Cambridge University Press 2000).]See also Nonlinear Diffusion Equations
& Reaction-Diffusion Systems. Topological degree theory for maps between Banach
spaces provides a means of estimating the number of solutions of an
equation, and has connections with fixed-point theory. It can be seen as an infinite-dimensional generalization of the
winding number of a curve in the complex plane. In examples where one
solution of a problem is easily spotted, degree theory can often be used to prove existence of a second, nontrivial,
solution. Current interests include the development of a degree
theory tailored to Hamiltonian systems, and applications to water waves. Professors Fraenkel, Toland and Vassiliev were awarded
the Senior Whitehead Prize (1989), the Senior Berwick Prize (2000) and
Whitehead Prize (1993), respectively, of the London Mathematical Society for work in analysis and differential equations. | ||||
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