739 research outputs found
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 and is controlled by an internal cell pressure . We
assume that the tumor occupies a strip-like domain with a fixed boundary at
and a free boundary , where is a -periodic
function. First, we prove the existence of solutions 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 , we let in the right half-plane
and in the left half-plane , where is the usual matrix describing
rotation of the coordinates in by an angle . As a main result,
it is shown that spectral gaps of the periodic Schr\"odinger operator fill with spectrum of as . 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 -component nonlinear evolutionary PDE which includes the
-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 -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 --equation on the
diffeomorphism group of the torus in this context.Comment: 14 page
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