Motivated by the study of fibre dynamics in the carding machine, a continuum model for the motion of a medium composed of fibres is derived under the assumption that the dominant forces are due to fibre-fibre interactions and that the material is in tension. To characterise the material we include the averaged values of density and velocity and introduce variables to describe the mean direction, alignment and entanglement. We assume that the bulk stress of the material depends on the density, entanglement, degree of alignment, average direction and shear-rates. A kinematic equation for average direction and two proposed heuristic laws for the evolution of entanglement and degree of alignment are given to close the system. Extensional and shearing simulations are in good qualitative agreement with experimental results.

An optimization problem for the fundamental eigenvalue $\lam_0$ of the Laplacian in a planar simply-connected domain that contains $N$ small identically-shaped holes, each of radius $\eps\ll 1$, is considered. The boundary condition on the domain is assumed to be of Neumann type, and a Dirichlet condition is imposed on the boundary of each of the holes. As an application, the reciprocal of the fundamental eigenvalue $\lam_0$ is proportional to the expected lifetime for Brownian motion in a domain with a reflecting boundary that contains $N$ small traps. For small hole radii $\eps$, a two-term asymptotic expansion for $\lam_0$ is derived in terms of certain properties of the Neumann Green's function for the Laplacian. Only the second term in this expansion depends on the locations $x_{i}$, for $i=1,\ldots,N$, of the small holes. For the unit disk, ring-type configurations of holes are constructed to optimize this term with respect to the hole locations. The results yield hole configurations that asymptotically optimize $\lam_0$. For a class of symmetric dumbbell-shaped domains containing exactly one hole, it is shown that there is a unique hole location that maximizes $\lam_0$. For an asymmetric dumbbell-shaped domain, it is shown that there can be two hole locations that locally maximize $\lam_0$. This optimization problem is found to be directly related to an oxygen transport problem in skeletal muscle tissue, and to determining equilibrium locations of spikes to the Gierer–Meinhardt reaction-diffusion model. It is also closely related to the problem of determining equilibrium vortex configurations within the context of the Ginzburg–Landau theory of superconductivity.

We consider the following Gray–Scott model in $B_R (0) = \{x: |x|\lt R\} \subset \R^N, N=2,3$: \[ \left\{ \begin{array}{@{}ll} v_t= \ep^2 \oldDelta v- v+ A v^2 u &\mbox{ in } B_R (0),\\ \tau u_t=\oldDelta u+1-u - v^2 u & \mbox{ in } B_R(0), \\ u, v\gt 0;~~ \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu} =0 &\hspace*{3pt}\mbox{on} \ \partial B_R(0) \end{array} \right. \] where $\ep>0$ is a small parameter. We assume that $A= \skew{7}\hat{A} \ep^{\frac{1}{2}}$. For each $\skew{7}\hat{A} < +\infty$ and $R<\infty$, we construct ring-like solutions which concentrate on an $(N-1)$-dimensional sphere for the stationary system for all sufficiently small $\ep$. More precisely, it is proved the above problem has a radially symmetric steady state solution $(v_{\ep,R}, u_{\ep, R})$ with the property that $v_{\ep,R} (r) \to 0 $ in $\R^N \backslash \{ r \not = r_0 \}$ for some $ r_0 \in (0, R)$. Then we show that for $N=2$ such solutions are unstable with respect to angular fluctuations of the type $ \Phi (r) e^{ \sqrt{-1} m \theta}$ for some $m$. A relation between $\skew{7}\hat{A}$ and the minimal mode $m$ is given. Similar results are also obtained when $\Om=\R^N$ or $\Om= B_{R_2} (0) \backslash B_{R_1} (0)$ or $\Om= \R^N \backslash B_R (0)$.

For a given scalar partial differential equation (PDE), a potential variable can be introduced through a conservation law. Such a conservation law yields an equivalent system (potential system) of PDEs with the given dependent variable and the potential variable as its dependent variables. Often there is also another equivalent scalar PDE (potential equation) with the potential variable as its dependent variable. The Nonclassical Method for obtaining solutions of PDEs is a generalization of the Classical Method for obtaining invariant solutions from point symmetries admitted by a given PDE. As a prototypical example, the nonlinear heat conduction equation is used to demonstrate that the Nonclassical Method applied to a potential equation can yield new solutions (nonclassical potential solutions) of a given PDE that are unobtainable as invariant solutions from admitted point symmetries of the given PDE, a related potential system or the potential equation, or from nonclassical solutions generated by applying the Nonclassical Method ($\tau{\equiv} 1$) to the given scalar PDE.

The aim of this paper is to provide a survey on the recent development in level set methods in inverse problems and optimal design. We give introductions on the general features of such problems involving geometries and on the general framework of the level set method. In subsequent parts we discuss shape sensitivity analysis and its relation to level set methods, various approaches on constructing optimization algorithms based on the level set approach, and special tools needed for the application of level set based optimization methods to ill-posed problems. Furthermore, we provide a review on numerical methods important in this context, and give an overview of applications treated with level set methods. Finally, we provide a discussion of the most challenging and interesting open problems in this field, that might be of interest for scientists who plan to start future research in this field.