IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

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

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

/pdf/nonlinear-partial-differential-equations-with-applications-204hhsvanv.pdf

Nonlinear partial differential equations with applications

We consider the boundary value problem $
\Delta u + \varepsilon ^{2} k{\left( x \right)}e^{u} = 0$
 in a bounded, smooth domain $\Omega$
 in $
\mathbb{R}^{{\text{2}}} 
$

 with homogeneous Dirichlet boundary conditions. Here $$
\varepsilon > 0,k(x)
$$

 is a non-negative, not identically zero function. We find conditions under which there exists a solution $
u_{\varepsilon } 
$
 which blows up at exactly m points as $
\varepsilon \to 0
$
 and satisfies $
\varepsilon ^{2} {\int_\Omega {ke^{{u_{\varepsilon } }} \to 8m\pi } }%
$

 . In particular, we find that if $k\in C^2(\bar\Omega)$
 , $
\inf _{\Omega } k > 0
$
 and $\Omega$
 is not simply connected then such a solution exists for any given $m \ge 1$

/pdf/singular-limits-in-liouville-type-equations-4a2a66zgxn.pdf

Singular limits in liouville-type equations

has at least one solution when q < N+2 N−2 for any smooth bounded domain Ω. On the contrary, when q is critical or supercritical the existence of solutions to problem (1.2) depends strongly on the shape of the domainΩ. Indeed, if q ≥ N+2 N−2 Pohozaev’s identity [18] gives that problem (1.2) has no solution ifΩ is star-shaped. On the other hand, if q = N+2 N−2 , problem (1.2) has at least one solution when Ω is a symmetric annulus, see Kazdan-Warner [15], or when Ω has a “small hole”, see Coron [9]. In a remarkable work [3], Bahri and Coron generalize the previous results, by proving that if q = N+2 N−2 and if some homology group of Ω with coefficients in Z2 is nontrivial, then problem (1.2) has a solution.

/pdf/two-bubblesolutions-in-the-super-critical-bahri-coron-s-41iuke9ptr.pdf

Two-bubblesolutions in the super-critical Bahri-Coron's problem

http://www.cmls.polytechnique.fr/perso/pacard.frank/Publications/64.pdf

Large energy entire solutions for the yamabe equation

The main purpose of this paper is to construct families of positive solutions for the equation formula math. which blow-up and concentrate in k ≥ 1 different points of Q as e goes to 0. We exhibit some examples of contractible domains where a large number of solutions exists.

ultispike solutions for a nonlinearelliptic problem involving critical Sobolev exponent

http://www.mat.uniroma3.it/users/esposito/Papers/JDE1.pdf

Monica Musso

Papers

Singular limits in liouville-type equations

Two-bubblesolutions in the super-critical Bahri-Coron's problem

Large energy entire solutions for the yamabe equation

ultispike solutions for a nonlinearelliptic problem involving critical Sobolev exponent

Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent