On the stability of a forward-backward heat equation
with L. Boulton and M. Marletta, Integral Equations Operator Theory, vol. 73 (2012), no. 2, pp. 195-216.
In this paper we examine spectral properties of a family of periodic singular Sturm-Liouville problems which are highly non-self-adjoint but have purely real spectrum. The problem originated from the study of the lubrication approximation of a viscous fluid film in the inner surface of a rotating cylinder and has received a substantial amount of attention in recent years. Our main focus will be the determination of Schatten class inclusions for the resolvent operator and regularity properties of the associated evolution equation.
Weighted Lp boundedness of pseudodifferential operators and applications
with N. Michalowski and W. Staubach, Canad. Math. Bull., published electronically, June 2011 pp.1-16.
In this paper we prove weighted norm inequalities with weights in the Ap classes, for pseudodifferential operators with symbols in the class Smρ,δ
which fall outside the scope of Calderón-Zygmund theory. This paper can
be viewed as a prelude to Michalowski-Rule-Staubach (below), in that
the main result here is a particular case of the results in that paper.
However, the techniques used here are rooted in the theory of smooth
Elliptic equations in the plane satisfying a Carleson measure condition
with M. Dindoš, Rev. Mat. Iberoam., vol. 26 (2010), no. 3, pp. 1013-1034.
Once again we study divergence form elliptic operators which are not assumed to be symmetric in domains in R2 above the graph of a Lipschitz function. However, here we assume the coefficients satisfy a Carleson measure condition. Using a new technique of introducing an auxiliary equation, we can prove that the Neumann and regularity problems are solvable with data in Lp for some p>1 provided the Carleson measure norm is sufficiently small.
Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes
with N. Michalowski and W. Staubach, J. Funct. Anal., vol. 258, 12, pp. 4183-4209.
We study pseudodifferential operators which are only
assumed to be measurable in the spatial variable. We give conditions
under which these operators are bounded on weighted Lp
with weights in the Muckenhoupt classes. Some of the results are shown
to be sharp with respect to these hypotheses, however, for operators of a
particular form, the hypotheses can be weakened. As an application of
these weighted boundedness results we show that the commutators of these
operators with functions of bounded mean oscillation are bounded in Lp.
The regularity and Neumann problem for non-symmetric elliptic operators
with C.E. Kenig, Trans. Amer. Math. Soc., vol. 361 (2009), pp. 125-160.
We study divergence form elliptic operators which are not assumed to be symmetric in domains in R2 above the graph of a Lipschitz function. Under the assumption that the coefficients of the operator are independent of the vertical direction and measurable in the horizontal, we prove that the Neumann and regularity problems are solvable with data in Lp for some p>1. This is done via an application of David and Journé's T(b) Theorem and the extra regularity properties of solutions in R2.
Non-symmetric elliptic operators on bounded Lipschitz domains in the plane,
Electron. J. Diff. Eqns., vol. 2007 (2007), no. 144, pp. 1-8.
This paper extends the results of Kenig-Rule (above) to include bounded Lipschitz domains in R2.
We modify the arguments in Kenig-Rule to enable us to prove the
boundedness of layer potentials in the more general context required for