Survey paper: Set invariance in control

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Introduction Suggestions for the readerSuggestions for the reader Fredholm mappings of index zero and linear boundary value problems Degree theory for some classes of mappings Duality theorems for several fixed point operators associated to periodic problems for ordinary differential equations Existence theorems for equations in normed spaces Boundary value problems for second order nonlinear vector differential equations Periodic solutions of ordinary differential equations with one-sided growth restrictions Bound sets for functional differential equations The index of isolated zeros of some mappings Bifurcation theory Periodic solutions of autonomous ordinary differential equations around an equilibrium References.

Topological degree methods in nonlinear boundary value problems

http://personal.strath.ac.uk/x.mao/pdffiles/explo_pop.pdf

Environmental Brownian noise suppresses explosions in population dynamics

We present an inventory and analysis of discussions of ecological stability, considering 163 definitions of 70 different stability concepts. Our aim is to derive a strategy that can help to dispel the existing ”confusion of tongues” on the subject of ”stability” and prevent its future recurrence. The strategy consists of three questions that should be kept in mind when communicating about stability properties. These three questions should overcome the three main sources of confusion in terminology. Firstly, which stability properties are being addressed in the stability statement? Our analysis shows that the general term ”stability” is so ambiguous as to be useless.It can be replaced by the stability properties ”staying essentially unchanged” (constancy), ”returning to the reference state (or dynamic) after a temporary disturbance” (resilience), and ”persistence through time of an ecological system” (persistence). Second, to what ecological situation does the statement refer? An ecological situation is defined by a set of features that, taken as a whole, determine the domain of validity of a stability statement. The six most important features form the ”ecological checklist”, which serves to classify ecological situations and thereby provides a system of coordinates for communication. The six points are: variable of interest, level of description, reference state, disturbance, spatial scale and temporal scale. Thirdly, is the statement anchored in the situation in question, or is there unacceptable generalisation by inferring ”stability” of the whole system from a certain stability property in a certain ecological ecological situation? This question separates the scientifically valuable content of a statement from the desire for general statements which is often projected through stability statements.

Babel, or the ecological stability discussions: an inventory and analysis of terminology and a guide for avoiding confusion.

Variational techniques are applied to prove the existence of standing wave solutions for quasilinear Schrodinger equations containing strongly singular nonlinearities which include derivatives of the second order. Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Direct methods of the calculus of variations and minimax methods like the Mountain Pass Theorem are used. The difficulties introduced by the nonconvex functional $\Phi(u)=\int |\nabla u|^2 u^2$ are substantially different from the semilinear case.

On the existence of soliton solutions to quasilinear Schrödinger equations

A basic question in mathematical biology concerns the long-term survival of each component, which might typically be a population in an ecological context, of a system of interacting components. Many criteria have been used to define the notion of long-term survival. We consider here the subject of permanence, i.e., the study of the long-term survival of each species in a set of populations. These situations may often be modeled successfully by dynamical systems and have led to the development of some interesting mathematical techniques and results. Our intention here is to describe these and to consider their application to several of the most frequently used models occurring in mathematical biology. We particularly wish to include and cover those models leading to problems that are essentially infinite dimensional, for example reaction-diffusion equations, and to make the discussion accessible to a wide audience, we include a chapter outlining the fundamental theory of these.

Permanence and the dynamics of biological systems.

https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1771&context=hmc_fac_pub

The Liouville–Bratu–Gelfand Problem for Radial Operators

We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problem 
$$(D)\quad \left \{\begin{array}{rcll} -{\rm div} \left (a(|\nabla u(x)|)\nabla u(x)\right )& =& g(x,u), & \mbox{in} \Omega u& = &0, & \mbox{on} \partial\Omega, \end{array} \right .$$
 where 
$\Omega $
 is a bounded domain in 
${\mathbb R}^N$
 , 
$g\in C(\overline{\Omega}\times\mathbb R,\mathbb R)$
 , and the function 
$\phi(s)= sa(|s|)$
 is an increasing homeomorphism from 
${\mathbb R}$
 onto 
${\mathbb R}$
 . Under appropriate conditions on 
$\phi$
 , 
$g$
 , and the Orlicz-Sobolev conjugate 
$\Phi_*$
 of 
$\Phi(s)=\int_0^s\phi(t) dt,$
 (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers), we obtain the existence of nontrivial solutions which are of mountain pass type.

Klaus Schmitt

Papers

On the existence of soliton solutions to quasilinear Schrödinger equations

Permanence and the dynamics of biological systems.

The Liouville–Bratu–Gelfand Problem for Radial Operators

Mountain pass type solutions for quasilinear elliptic equations

On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting