Abstract. We have obtained the best possible conditions for removable singularity at the point for solutions of quasilinear parabolic equations of divergent form. Cases of interior singular point (x0, t0) ∈ QT ⊂ R n+1 we have established a removability result for a solution u(x, t) ∈ V 2,p loc (QT )\ (x0, t0) under a condition u(x, t) = o([|x−x0|+ |t−t0| 1 p+n(p−2) ]). As a particular case we have a precise removability condition for p-Laplacian evolution equation. The proof is based on a new approach connected with point-wise estimates of solutions in puncturated domains.

For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit.

We study initial-boundary value problems for elliptic-parabolic systems of nonlinear partial differential equations describing drift-diffusion- reaction processes of electrically charged species in N-dimensional bounded Lipschitzian domains. We include Fermi-Dirac statistics and admit nonsmooth material coefficients. We prove existence and uniqueness of bounded global solutions.

Long-range order is proved to exist for lattice linear oscillator systems with ferromagnetic potential energy containing a term with strong nearest-neighbor (n-n) quadratic pair potential. A contour bound and a generalized Peierls argument are used in the proof.

We study the problem of non-existence of positive solutions to the elliptic inequalities involving quasilinear operators of the type −divA(x,u,∇u)≥|x| s u p , in the exterior domains in ℝ N , N≥3, p>1.

We prove a priori estimates in \( L^2(0,T;W^{1,2}(\Omega)) \) and \( L^{\infty}(Q_T) \), existence and uniqueness of solutions to Cauchy-Dirichlet problems for elliptic-parabolic systems\( \frac {\partial \sigma(u)}{\partial t} - \sum\limits_{i=1}^n \frac {\partial}{\partial x_i} \left\{\rho(u) b_i \left (t,x,\frac {\partial (u-v)}{\partial x} \right) \right\} + a (t,x,v,u) = 0,\\- \sum\limits_{i=1}^n \frac {\partial}{\partial x_i} \left[ \kappa(x) \frac{\partial v}{\partial x_i} \right ] + \sigma(u) = f (t,x), \;(t,x) \in Q_T = (0,T) \times \Omega, \)where \( \rho(u) = \frac {\partial \sigma(u)}{\partial u} \). Systems of such form arise as mathematical models of various applied problems, for instance, electron transport processes in semiconductors. Our basic assumption is that \( \log \rho(u) \) is concave. Such assumption is natural in view of drift-diffusion models, where \( \sigma \) has to be specified as a probability distribution function like a Fermi integral and u resp. v have to be interpreted as chemical resp. electrostatic potential.

We study the behavior of eigenvalues and eigenfunctions of the Dirichlet problem for nonlinear elliptic second-order equations in domains with fine-grain boundary.

The convergence of polymer cluster expansions for correlation functions of general Gibbs oscillator-type systems and related nonequilibrium systems of Brownian oscillators is established. The initial states for the latter are Gibbsian. It is proved that the sequence of the constructed correlation functions of the nonequilibrium system is a generalized solution of a diffusion BBGKY-type hierarchy.

We obtained best possible conditions for removable singularity at the point for solutions of nonlinear elliptic second order equations of divergent form. Well known Serrin's results guarantee the removability of the singularity at the point x(0) if u(x) is an element of W-loc(1,m)(Omega\{x(0)}) and u(x) = o(\x - x(0)\((m-n)/(m-1)+delta)) for m < n or u(x) = o(\log\x - x(0)parallel to(1-delta)) if m = n with some positive number delta. We established analogous result with delta = 0. The proof is based on new approach connected with point-wise estimates of singular solutions.

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W m 1(Ω s ) → [W m 1(Ω s )]* in a sequence of perforated domains Ω s ⊂ Ω. Under a certain condition imposed on the local capacity of the set Ω \ Ω s , we prove the following principle of compensated compactness: \({\mathop {\lim }\limits_{s \to \infty }} \left\langle {Ar_s ,z_s } \right\rangle = 0\), where rs(x) and zs(x) are sequences weakly convergent in W m 1(Ω) and such that rs(x) is an analog of a corrector for a homogenization problem and zs(x) is an arbitrary sequence from \({\mathop {W_m^1 }\limits^ \circ} (\Omega _s)\) whose weak limit is equal to zero.

Let $X$ be a real separable reflexive Banach space with dual space $X^*.$ Assume that the operator $A:X\supset\mathcal{D}(A)\to X^*$ is such that $L\subset \mathcal{D}(A)$ and $L=X,$ where $L$ is a subspace of $X.$ It is shown that it is possible to define a topological degree for such operators $A$ that satisfy, mainly, Condition $(S_+)_{0,L}.$ It is also shown that a topological degree can be defined for operators of the type $M+A,$ where $M+A: X\supset\mathcal{D}(M+A)\to X^*, L\subset\mathcal{D}(M+A),$ and $ L=X.$ Here, $X$ is not necessarily separable and $M$ satisfies a variant of the maximal monotonicity condition (with respect to the space $L$) as well as an approximation condition. The operator $A$ satisfies, mainly, analogues of the quasi-boundedness condition and the $(S_+)$ condition (with respect to the operator $M).$ Properties of these degrees are studied and applications are given for nonlinear Dirichlet elliptic problems of the type $$ \sum_{i=1}^n\frac{\partial}{\partial x_i}\left\{\rho^2(u)\frac{\partial u}{\partial x_i}+ a_i\left(x,u,\frac{\partial u}{\partial x}\right)\right\}=\sum_{i=1}^n\frac{\partial}{\partial x_i}f_i(x), $$ as well as Cauchy-Dirichlet parabolic problems of the type $$ \frac{\partial u}{\partial t}-\sum_{i=1}^n\frac{\partial}{\partial x_i}a_i\left(x,t,u,\frac{\partial u}{\partial x}\right) +\rho(x,t,u)=\sum_{i=1}^n\frac{\partial}{\partial x_i}f_i(x,t). $$

The eminent Soviet geometer Aleksei Vasil'evich Pogorelov was sixty years old on 3 March 1979. He comes from a working-class family in Koroch, a town in the Belgorod province, and received his secondary education in Kharkhov. When he was still at school, his brilliant abilities were noticed by the organizers of the Mathematical Olympiads: he had a highly developed geometrical imagination, a rich mathematical intuition and a rigorous mind for logical analysis. He invariably won first prizes in the local and national Olympiad competitions. In 1973 he entered the Mathematics Department of the Physics and Mathematics Faculty at the University of Kharkhov. He did well in all subjects, but he was particularly interested in geometry and formed close ties with the people attached to the geometry department, which at that time was headed by D. M. Sintsov. Pogorelov was in his final year when the war started. He joined the Red Army and served in the Air Force Academy in Moscow until 1945, when he started work as a designer at the Zhukovskii Central Institute of Aerohydrodynamics. At the same time he was taking a postgraduate correspondence course at the Institute of Mathematics of the University of Moscow under the supervision of N. V. Efimov. It was here where he first met A. D. Alexandrov. He received his Ph.D. in 1947 after which he started working at the Institute of Mathematics and Mechanics at the University of Kharkhov. In 1948, after only one year there, he presented his doctoral thesis and was put in charge of the institute's department of geometry. He has since been head of the geometry departments at the University of Kharkhov (1950—1959), the Institute of Mathematics attached to the Ukrainian branch of the Academy of Science (1959—1960) and, since 1960, at the Physico-Technical Low Temperature Institute, also part of the Ukrainian branch of the Academy of Sciences. Pogorelov was elected a Corresponding Member of the Ukranian SSR Academy of Sciences in 1957, became a full Member in 1960, and then a Member of the USSR Academy of Sciences in 1977. Since 1978 he has also been a Member of the Praesidium of the Ukranian SSR Academy of Sciences

We study the problem of averaging of Dirichlet problems for degenerate nonlinear elliptic equations of the second order in domains with fine-grained boundary under the condition that the weight function belongs to a certain Muckenhoupt class. We prove a pointwise estimate for solutions of the model degenerate nonlinear problem. The averaged boundary-value problem is constructed under new structural conditions for a perforated domain. In particular, we do not assume that the diameters of cavities are small as compared with the distances between them.

In a domain D=Ω\E∈ R n , we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is W p m (D)⩜W q 1 (D), q>mp, and estimate a solution u(x) of this equation satisfying the condition u(x)−kf(x)∈W p m (D)⩜W q 1 (D), where k∈R 1, f(x)∈ C 0 ∞ (Ω), and f(x)=1 for x∈F. We establish a pointwise estimate for u(x) in terms of the higher-order capacity of the set F and the distance from the point x to the set F.

Finite volume grand canonical correlation functions of nonequilibrium systems of d-dimensional Brownian particles, interacting through a regular (long-range) pair potential with integrable first partial derivatives, are expressed in terms of the expectation values of a Gaussian random field. The initial correlation functions coincide with the Gibbs correlation functions corresponding to a more general pair long-range potential. Nonequilibrium Euclidean action is introduced, satisfying a thermodynamic stability property.