Partial Differential Equations Commons^™

The Dual Of The Compressed Shift, M. C. Câmara, William T. Ross

Department of Math & Statistics Faculty Publications

For an inner function u, we discuss the dual operator for the compressed shift P_uS∣_Ku, where K_u is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.

Partially Isometric Matrices: A Brief And Selective Survey, Stephan Ramon Garcia, Matthew Okubo Patterson, William T. Ross

Department of Math & Statistics Faculty Publications

We survey a variety of results about partially isometric matrices. We focus primarily on results that are distinctly finite-dimensional. For example, we cover a recent solution to the similarity problem for partial isometries. We also discuss the unitary similarity problem and several other results.

Inner Vectors For Toeplitz Operators, Raymond Cheng, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we survey and bring together several approaches to obtaining inner functions for Toeplitz operators. These approaches include the classical definition, the Wold decomposition, the operator-valued Poisson Integral, and Clark measures. We then extend these notions somewhat to inner functions on model spaces. Along the way we present some novel examples.

Clark Measures And A Theorem Of Ritt, Isabelle Chalendar, Pamela Gorkin, Jonathan R. Partington, William T. Ross

Department of Math & Statistics Faculty Publications

We determine when a finite Blaschke product B can be written, in a non-trivial way, as a composition of two finite Blaschke products (Ritt's problem) in terms of the Clark measure for B. Our tools involve the numerical range of compressed shift operators and the geometry of certain polygons circumscribing the numerical range of the relevant operator. As a consequence of our results, we can determine, in terms of Clark measures, when two finite Blaschke products commute.

On The Flow Of Non-Axisymmetric Perturbations Of Cylinders Via Surface Diffusion, Jeremy Lecrone, Gieri Simonett

Department of Math & Statistics Faculty Publications

We study the surface diffusion flow acting on a class of general (non--axisymmetric) perturbations of cylinders Cr in IR3. Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially--unbounded) surfaces defined over Cr via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that Cr is normally stable with respect to 2π--axially--periodic perturbations if the radius r>1,and unstable if 0

On Quasilinear Parabolic Evolution Equations In Weighted Lp-Spaces Ii, Jeremy Lecrone, Mathias Wilke, Jan Prüss

Department of Math & Statistics Faculty Publications

Our study of abstract quasi-linear parabolic problems in time-weighted L_p-spaces, begun in 2010, is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to reaction-diffusion problems, including Maxwell-Stefan diffusion, and to geometric evolution equations like the surface-diffusion flow or the Willmore flow. The method presented here will be applicable to other parabolic systems, including free boundary problems.

Uniqueness For A Boundary Identification Problem In Thermal Imaging, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

An inverse problem for an initial-boundary value problem is considered. The goal is to determine an unknown portion of the boundary of a region in ℝⁿ from measurements of Cauchy data on a known portion of the boundary. The dynamics in the interior of the region are governed by a differential operator of parabolic type. Utilizing a unique continuation result for evolution operators, along with the method of eigenfunction expansions, it is shown that uniqueness holds for a large and physically reasonable class of Cauchy data pairs.

An Inverse Problem In Thermal Imaging, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

This paper examines uniqueness and stability results for an inverse problem in thermal imaging. The goal is to identify an unknown boundary of an object by applying a heat flux and measuring the induced temperature on the boundary of the sample. The problem is studied in both the case in which one has data at every point on the boundary of the region and the case in which only finitely many measurements are available. An inversion procedure is developed and used to study the stability of the inverse problem for various experimental configurations.

A Convergent Reconstruction Method For An Elliptic Operator In Potential Form, Lester Caudill

Department of Math & Statistics Faculty Publications

We investigate the problem of recovering a potential q(x) in the equation -∆u + q(x)u = 0 from overspecified boundary data on the unit square in R². The potential is characterized as a fixed point of a nonlinear operator, which is shown to be a contraction on a ball in C^∝. Uniqueness of q(x) follows, as does convergence of the resulting recovery scheme. Numerical examples, demonstrating the performance of the algorithm, are presented.

A Direct Method For The Inversion Of Physical Systems, Lester Caudill, Herschel Rabitz, Attila Askar

Department of Math & Statistics Faculty Publications

A general algorithm for the direct inversion of data to yield unknown functions entering physical systems is presented. Of particular interest are linear and non-linear dynamical systems. The potential broad applicability of this method is examined in the context of a number of coefficient-recovery problems for partial differential equations. Stability issues are addressed and a stabilization approach, based on inverse asymptotic tracking, is proposed. Numerical examples for a simple illustration are presented, demonstrating the effectiveness of the algorithm.

Determination Of A Potential From Cauchy Data: Uniqueness And Distinguishability, Lester Caudill

Department of Math & Statistics Faculty Publications

The problem of recovering a potential q(y) in the differential equation:

−∆u + q(y)u = 0 (x,y) &∈ (0, 1) × (0,1)
u(0, y) = u(1, y) = u(x, 0) = 0
u(x, 1) = f(x), u_y(x, 1) = g(x)

is investigated. The method of separation of variables reduces the recovery of q(y) to a non-standard inverse Sturm-Liouville problem. Employing asymptotic techniques and integral operators of Gel'fand-Levitan type, it is shown that, under appropriate conditions on the Cauchy pair (f, g ), q(y) is uniquely determined, in a local sense, up to its mean. We characterize …

On The Construction Of A Potential From Cauchy Data, Lester Caudill, Bruce D. Lowe

Department of Math & Statistics Faculty Publications

We investigate the problem of recovering a potential q(y)in the differential equation:

-∆u+q(y)u = 0, (x, y) ∈ (0, 1) x (0, 1),
u(0, y) = u(1, y) = u(x, 0) = 0,
u(x, 1) = f(x), u_y(x, 1) = g(x).

The method of separation of variables reduces the recovery of q(y) to a nonstandard inverse Sturm-Liouville problem. An asymptotic formula is developed that suggests that under appropriate conditions on the Cauchy pair (f, g), q(y) is uniquely determined up to the mean. Moreover, the recovery of …