Abstracts

Partial Differential Equations

An intrinsic description of the trace spaces H^s(∂G) and classes of harmonic functions on G.
by
Giles Auchmuty
University of Houston

This talk will outline an characterization of the Hilbert trace spaces H^s(∂G) on bounded regions G in Rⁿ with minimal boundary regularity. The description uses a spectral characterization in terms of the Steklov eigenfunctions of the Laplacian on the region. This leads to corresponding Hilbert spaces of real harmonic functions on the regions. It is shown that these spaces are reproducing kernel Hilbert spaces with respect to a natural inner product. Representation theorems for the solutions of Dirichlet, Robin and Neumann boundary value problems in these spaces are described.

Date received: October 31, 2008

Inverse positivity for general Robin problems on Lipschitz domains
by
Daniel Daners
The University of Sydney

We prove that elliptic boundary value problems in divergence form can be written in many equivalent forms. This is used to prove regularity properties and maximum principles for problems with Robin boundary conditions with negative or indefinite boundary coefficient on Lipschitz domains. We do this by rewriting such problem as a problem with positive boundary coefficient. We finally show that such a result cannot be proved for domains with an outward pointing cusp.

Date received: November 2, 2008

Convergence and thresholds in nonlinear diffusion problems
by
Yihong Du
University of New England
Coauthors: Hiroshi Matano

We study the Cauchy problem

ut=uxx+f(u) (t > 0,   x ∈ R1),    u(0, x)=u0(x) (x ∈ R1),

where f(u) is a locally Lipschitz continuous function satisfying f(0)=0. We show that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as t→∞. Moreover the limit is either a constant or a symmetrically decreasing stationary solution. We also consider the special case where f is a bistable nonlinearity and the case where f is a combustion type nonlinearity. Examining the behavior of a parameter-dependent solution u_l, we show the existence of a sharp threshold between extinction (namely, convergence to 0) and propagation (namely, convergence to 1). The result holds even if f has a jumping discontinuity at u=1.

Date received: October 19, 2008

Sectorial forms and degenerate operators
by
Tom ter Elst
University of Auckland
Coauthors: Wolfgang Arendt

In the theory of sectorial forms and holomorphic semigroups a basic assumption is that the form is closed, or at least closable. This is a nasty difficult condition. In a recent paper with Wolfgang Arendt we proved that one can associate in a natural way a holomorphic semigroup generator to any sectorial form, even if it is not closable. Thus one can forget closability. This opens the door to consider complex degenerate elliptic differential operators without demanding that they are symmetric or strongly elliptic. In the talk we present several examples and applications.

Date received: October 23, 2008

Direct " Delay" reductions of the Toda equation
by
Nalini Joshi
School of Mathematics and Statistics, The University of Sydney

A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painlevé equations. The Lax pair associated to this equation is obtained, also by reduction.

Paper reference: arXiv:0810.5581

Date received: November 1, 2008

Fully nonlinear curvature flows of nonconvex hypersurfaces
by
James McCoy
University of Wollongong

In this talk I consider a natural class of fully nonlinear curvature flows with closed, compact initial data that is not necessarily convex. I will show that some fundamental behaviour of solutions to these curvature flows is analogous to the case of the mean curvature flow, including the result that the only smooth, compact self-similar shrinking solutions of positive speed are shrinking spheres.

Date received: September 24, 2008

An abstract approach to domain perturbation for parabolic equations
by
Parinya Sa Ngiamsunthorn
University of Sydney

Let V be a reflexive Banach space. Suppose K_n, n ≥ 1 and K are closed and convex subsets of V. We show that Mosco convergence of K_n to K is equivalent to Mosco convergence of L²((0, T), K_n) to L²((0, T), K), where L²((0, T), K_n) consists of all function u ∈ L²((0, T), V) with u(t) ∈ K_n a.e. t ∈ (0, T). An application in domain perturbation for parabolic equations will be discussed.

Date received: October 22, 2008

Analysis of degenerate elliptic operators of Grusin type
by
Adam Sikora
Australian National University
Coauthors: Joint work with Derek W. Robinson

We analyze degenerate, second-order, elliptic operators H in divergence form on L₂(Rⁿ×R^m). We assume the coefficients are real symmetric and a₁H_d ≥ H ≥ a₂H_d for some a₁, a₂ > 0 where

Hd=-∇x1 cd1, d'1(x1) ∇x1-cd2, d'2(x1) ∇x22      .

Here x₁ ∈ Rⁿ, x₂ ∈ R^m and c_{di, d'i} are positive measurable functions such that c_{di, d'i}(x) behaves like |x|^d_i as x→0 and |x|^d_i' as x→∞ with d₁, d₁' ∈ [0, 1〉 and d₂, d₂' ≥ 0.

Our principal results state that the submarkovian semigroup S_t=e^-tH is conservative and its kernel K_t satisfies bounds

0 ≤ Kt(x ;y) ≤ a (|B(x ;t1/2)| |B(y ;t1/2)|)-1/2

where |B(x ;r)| denotes the volume of the ball B(x ;r) centred at x with radius r measured with respect to the Riemannian distance associated with H. The proofs depend on detailed subelliptic estimations on H, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation

Paper reference: math.AP/0607584

Date received: November 3, 2008

Bifurcation for some non-Fréchet differentiable problems
by
Charles Stuart
EPFL, Lausanne, Switzerland

We consider some basic aspects of bifurcation theory in the context of maps that are differentiable in the sense of Hadamard. In finite dimensions this property is equivalent to Fréchet differentiability, but in infinite dimensions it is a weaker condition. The stationary nonlinear Schrödinger equation will be used to illustrate the general results.

Date received: October 3, 2008

Hypersurface L^p estimates for approximate eigenfunctions of a differential operator
by
Melissa Tacy
Australian National University

In this talk I will present L^p estimates for approximate eigenfunctions of a differential operator restricted to a hypersurface of a compact manifold. This proof, similar to Koch, Tataru and Zworski's proof of eigenfunction estimates over the whole manifold, exploits locality to transform the problem into one concerning evolution equations. Strichartz estimates are then used, with one spatial variable taking the place of time, to achieve the required estimates.

Date received: October 28, 2008

Convergence of anisotropically decaying solutions of a semilinear parabolic equation
by
Eiji Yanagida
Mathematical Institute, Tohoku University

We consider the Cauchy problem for a semilinear parabolic equation with a supercritical power nonlinearity. It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In this talk we consider solutions whose initial value decays in an anisotropic way. Then we show that the solution converges to a steady state which is explicitly determined by an average formula. The proof is given by using previous results on the global stability and quasi-convergence of solutions, self-similar solutions of the linearized equation around a singular steady state, and a comparison technique. This is a joint work with Peter Polacik of the University of Minnesota.

Date received: October 30, 2008