Analysis of a mathematical model for the growth of cancer cells

In this paper, a two-dimensional model for the growth of multi-layer tumors is presented. The model consists of a free boundary problem for the tumor cell membrane and the tumor is supposed to grow or shrink due to cell proliferation or cell dead. The growth process is caused by a diffusing nutrient concentration $\sigma$ and is controlled by an internal cell pressure $p$. We assume that the tumor occupies a strip-like domain with a fixed boundary at $y=0$ and a free boundary $y=\rho(x)$, where $\rho$ is a $2\pi$-periodic function. First, we prove the existence of solutions $(\sigma,p,\rho)$ and that the model allows for peculiar stationary solutions. As a main result we establish that these equilibrium points are locally asymptotically stable under small perturbations.Comment: 15 pages, 2 figure

Spectral Properties of Grain Boundaries at Small Angles of Rotation

We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential $V \colon \R^2 \to \R$, we let $V_\theta(x,y) = V(x,y)$ in the right half-plane $\{x \ge 0\}$ and $V_\theta = V \circ M_{-\theta}$ in the left half-plane $\{x < 0\}$, where $M_\theta \in \R^{2 \times 2}$ is the usual matrix describing rotation of the coordinates in $\R^2$ by an angle $\theta$. As a main result, it is shown that spectral gaps of the periodic Schr\"odinger operator $H_0 = -\Delta + V$ fill with spectrum of $R_\theta = -\Delta + V_\theta$ as $0 \ne \theta \to 0$. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.Comment: 22 pages, 3 figure

A note on multi-dimensional Camassa-Holm type systems on the torus

We present a $2n$-component nonlinear evolutionary PDE which includes the $n$-dimensional versions of the Camassa-Holm and the Hunter-Saxton systems as well as their partially averaged variations. Our goal is to apply Arnold's [V.I. Arnold, Sur la g\'eom\'etrie diff\'erentielle des groupes de Lie de dimension infinie et ses applications \`a l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 (1966) 319-361], [D.G. Ebin and J.E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92(2) (1970) 102-163] geometric formalism to this general equation in order to obtain results on well-posedness, conservation laws or stability of its solutions. Following the line of arguments of the paper [M. Kohlmann, The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus. J. Phys. A.: Math. Theor. 44 (2011) 465205 (17 pp.)] we present geometric aspects of a two-dimensional periodic $\mu$-$b$-equation on the diffeomorphism group of the torus in this context.Comment: 14 page