Nonlocal and multipoint boundary value problems for linear evolution equations

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as \emph{multipoint conditions}, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e.\ that it admits a unique solution. For the second order case, we also give criteria that guarantee well-posedness.Comment: 28 pages, 4 figure

The Dirichlet-to-Neumann map for the elliptic sine Gordon

We analyse the Dirichlet problem for the elliptic sine Gordon equation in the upper half plane. We express the solution $q(x,y)$ in terms of a Riemann-Hilbert problem whose jump matrix is uniquely defined by a certain function b(\la), \la\in\R, explicitly expressed in terms of the given Dirichlet data $g_0(x)=q(x,0)$ and the unknown Neumann boundary value $g_1(x)=q_y(x,0)$, where $g_0(x)$ and $g_1(x)$ are related via the global relation \{b(\la)=0, \la\geq 0\}. Furthermore, we show that the latter relation can be used to characterise the Dirichlet to Neumann map, i.e. to express $g_1(x)$ in terms of $g_0(x)$. It appears that this provides the first case that such a map is explicitly characterised for a nonlinear integrable {\em elliptic} PDE, as opposed to an {\em evolution} PDE

Two-point boundary value problems for linear evolution equations

We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive nite constants. We present a general method for identifying well-posed problems, as well as for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iqt + qxx = 0 and qt + qxxx = 0

Boundary value problems for the elliptic sine-Gordon equation in a semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2 by 2 matrix Riemann-Hilbert problem whose \jump matrix" depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however a major difficulty for this problem is the existence of non-integrable singularities of the function q_y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in terms of a 2 by 2 matrix Riemann-Hilbert problem whose jump matrix depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h. The determination of the function h remains open

Evolution equations on time-dependent intervals

We study initial boundary value problems for linear evolution partial differential equations (PDEs) posed on a time-dependent interval $l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterise the unknown boundary values in terms of the given initial and boundary conditions. As illustrative examples we consider the heat equation and the linear Schr\"{o}dinger equation. In the first case, the unknown Neumann boundary values are expressed in terms of the Dirichlet boundary values and of the initial value through the unique solution of a system of two linear integral equations with explicit kernels. In the second case, a similar result can be proved but only for a more restrictive class of boundary curves.

Sphincter-sparing surgery after preoperative radiotherapy for low rectal cancers: feasibility, oncologic results and quality of life outcomes

The present study assesses the choice of surgical procedure, oncologic results and quality of life (QOL) outcomes in a retrospective cohort of 53 patients with low-lying rectal cancers (within 6 cm of the anal verge) treated surgically following preoperative radiotherapy (RT, median dose 45 Gy) with or without concomitant 5-fluorouracil. QOL was assessed in 23 patients by using two questionnaires developed by the QOL Study Group of the European Organization for Research and Treatment of Cancer: EORTC QLQ-C30 and EORTC QLQ-CR38. After a median interval of 29 days from completion of RT, abdominoperineal resection (APR) was performed in 29 patients (55%), low anterior resection in 23 patients (20 with coloanal anastomosis) and transrectal excision in one patient. The 3-year actuarial overall survival and locoregional control rates were 71.4% and 77.5% respectively, with no differences observed between patients operated by APR or restorative procedures. For all scales of EORTC QLQ-C30 and EORTC QLQ-CR38, no significant differences in median scores were observed between the two surgical groups. Although patients having had APR tended to report a lower body image score (P = 0.12) and more sexual dysfunction in male patients, all APR patients tended to report better physical function, future perspective and global QOL. In conclusion, sphincter-sparing surgery after preoperative RT seems to be feasible, in routine practice, in a significant proportion of low rectal cancers without compromising the oncologic results. However, prospective studies are mandatory to confirm this finding and to clarify the putative QOL advantages of sphincter-conserving approaches. © 2000 Cancer Research Campaig