Table of contents

CONTENTS Chapter 1. The structure of singularities § 1. Examples § 2. The classes Σ^I § 3. The quadratic differential § 4. The local ring of a singularity and the Weierstrass preparation theorem Appendix. A proof of the Weierstrass preparation theorem Chapter 2. Deformations of singularities § 5. "Infinite-dimensional Lie groups" acting on "infinite-dimensional manifolds" § 6. The stability theorem § 7. Proof of convergence § 8. In the neighbourhood of an isolated critical point every analytic function is equivalent to a polynomial References

This paper is a survey of recent results on the solution of boundary value problems for quasilinear elliptic and parabolic equations of order 2m, of divergent form. The main results in this direction were first obtained in 1961 by Vishik, Browder, the author and others, and are presented in the first part of the paper. We also indicate the spaces in which the elliptic and parabolic operators induce homeomorphisms in the strongly elliptic case. When the variation of the operator is merely semibounded below, the Dirichlet problem is soluble for any right-hand side, though not uniquely. In the second part we present the work of several authors concerning the solution of operator equations in Banach spaces, among them Minty, Browder, Leray, Lions, Dubinskii, Pokhozhaev. The results are then applied to non-linear differential equations.

Let be a continuum (other than a single point) in the -plane not disconnecting the plane, a simply-connected domain containing . The class consists of those functions that are analytic in and satisfy the inequality

The author proves the following theorem:

Here is the -entropy of , and the -dimensional linear diameter of in the space of all functions continuous on . The norm on is

For the proof a basis is constructed in the space of functions holomorphic in ; it coincides with the Faber basis if is a level curve of . A fundamental part in this construction is played by a lemma which states that the domain can be mapped conformally into a domain , where is a level curve of . In the appendix, which is written by A. L. Levin and V. M. Tikhomirov, a similar theorem is proved (under additional assumptions) for the case when is multiply-connected and may consist of several continua.

CONTENTS Preface (V. M. Tikhomirov) Introduction § 1. The problem of bases. The main lemma § 2. Fundamental properties of the bases constructed § 3. Asymptotic theory of ε-entropy § 4. Estimates for n-dimensional diameters § 5. Some "extraneous" results Appendix. A. L. Levin and V. M. Tikhomirov, On a theorem of Erokhin References

These lectures contain a discussion of the fundamental concepts of statistical physics, such as the Gibbs distribution, correlation functions, magnitudes in thermodynamics, etc. Besides concepts that are familiar to physicists, the lectures introduce some new theorems and developments of a more mathematical character, namely, the limiting Gibbs distribution, van Hove's theorem, and equations for limiting correlation functions in lattice systems. The lecture on phase transitions of the first kind is the hub of the article; it contains the greater part of the known mathematical results concerning phase transitions of the first kind in lattice systems.