Yu. A. Dubinskii

Nonlinear elliptic and parabolic equations

Results of recent years are presented on the theory of nonlinear elliptic and parabolic equations of any order including equations of infinite order.

Some coercive problems for the system of Poisson equations

In the paper, two boundary value problems for the system of Poisson equations in three-dimensional domains are studied.

A Hardy-type inequality and its applications

We prove a Hardy-type inequality that provides a lower bound for the integral ∫0∞|f(r)|prp−1dr, p > 1. In the scale of classical Hardy inequalities, this integral corresponds to the value of the exponential parameter for which neither direct nor inverse Hardy inequalities hold. However, the problem of estimating this integral and its multidimensional generalization from below arises in some practical questions. These are, for example, the question of solvability of elliptic equations in the scale of Sobolev spaces in the whole Euclidean space ℝn, some questions in the theory of Sobolev spaces, hydrodynamic problems, etc. These questions are studied in the present paper.

High order nonlinear parabolic equations

The basic results and methods of the theory of high order nonlinear parabolic equations are described. In the first chapter boundary problems for quasilinear parabolic equations having divergent form are considered. In the second chapter nonlinear parabolic equations of general form are considered. Attention is mainly paid to methods of study of nonlinear parabolic problems. In particular, the methods of monotonicity and compactness, the method of a priori estimates, the functional-analytic method, etc. are described.

Decompositions of the Sobolev scale and gradient-divergence scale into the sum of solenoidal and potential subspaces

For the complete Sobolev scale and the gradient-divergence scale, decompositions into direct sums of solenoidal and potential subspaces are found. A smoothing property of solenoidal factorization is proved. Projectors onto the subspaces of solenoidal and potential functions are described.

Decompositions of the Sobolev-Clifford modules and nonlinear variational problems

We establish a general direct decomposition of modules and then, using this decomposition, prove representations of the Sobolev-Clifford modules as the sums of submodules of monogenic and comonogenic functions. We also show how the decompositions obtained can be applied to solving Stokes-type nonlinear variational problems.

The algebra of pseudodifferential operators with complex arguments and its applications

A number of results presently available in the theory of p/d operators with complex arguments is systematically presented in the survey, and their applications to partial differential equations are given.

On the choice of weight functions in Hardy-type inequalities

This paper presents a constructive description of all possible weighting factors in Hardytype inequalities. An exact description of these factors turned out to be possible within the framework of measures that are the sum of the regular Lebesgue-Stieltjes measure and the Dirac measure. As an application, multidimensional weighted inequalities in n , n > 1, of the Friedrichs and Poincare type are obtained.

The problem of continuation with least coanalytic deviation

The problem of continuing a function from the unit circle to the unit disk so that the continuation has the least deviation from the Sobolev subspace of analytic functions is considered. A mathematical model of this problem is constructed. It is proved that the problem is well-posed.

A nonpercolation problem for the stationary Navier-Stokes equations

The solvability of a nonpercolation boundary problem for the stationary Navier-Stokes equations is proved. The key points of the proof are analogues of the Friedrichs inequality and the de Rham theorem adequate for nonpercolation conditions.