The averaging method and the asymptotic behavior of solutions to differential inclusions

We prove one version of the first Bogolyubov theorem for differential inclusions with multivalued mappings that satisfy certain one-sided constraints. We study the dependence of solutions to differential inclusions on the parameters.     

Error estimation of approximate solutions to one inverse problem for a parabolic equation

This paper is devoted to solving the inverse boundary problem of the heat diagnostics by the projective regularization method. We obtain exact with respect to the order error estimates of the corresponding approximate solution.     

A certain boundary-value problem with shifts in a four-dimensional space

We consider a problem with shifts in boundary conditions for the Bianchi equation in a four-dimensional space. We establish sufficient conditions which allow one to evaluate its unique solution in terms of the Riemann function.     

Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions

Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂_ν u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L ^∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s| ^p ) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) =  s ^p then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of ∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential equation is of the form h(x, u) with h satisfying suitable growth conditions.     

The Neumann problem for nonlocal nonlinear diffusion equations

We study nonlocal diffusion models of the form

On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

A nonstationary group pursuit problem

We consider a linear nonstationary problem of conflict interaction of controlled objects, where the number of pursuers equals ν and the number of evaders equals µ. All participants are assumed to have equal dynamic abilities. The purpose of the pursuers is to catch all evaders, while the purpose of the latter is to avoid being caught for at least one of them. We establish sufficient solvability conditions for the local evasion problem.     

Existence of solutions and periodic solutions for nonlinear evolution inclusions

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.     

The Cauchy problem in Sobolev spaces for Dirac operators

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ? ? ⁿ with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function u in a Sobolev space H ^s (D), s ∈ ?, from its values on Γ and values Au in D, where Γ is an open connected subset of the boundary ?D. It is worth pointing out that we impose no assumptions about geometric properties of the domain D, except for its connectedness.     

Localization of the method of guiding functions in the problem about periodic solutions of differential inclusions

In this paper we consider the problem on the existence of forced oscillations in nonlinear objects governed by differential inclusions. We propose certain modifications of the methods of generalized and integral guiding functions.     

Multiplicity results for a Neumann problem withp-Laplacian and non-smooth potential

A multiplicity theorem for a non-smooth homogeneous Neumann problem withp-Laplacian is established through a locally Lipschitz continuous version of the Brézis-Nirenberg critical point result in presence of splitting. Some special cases are then pointed out.     

Positive solutions for second-order superlinear repulsive singular Neumann boundary value problems

In this paper we establish the multiplicity of positive solutions to second-order superlinear repulsive singular Neumann boundary value problems. It is proved that such a problem has at least two positive solutions under reasonable conditions. Our nonlinearity may be repulsive singular in its dependent variable and superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.      

Local Minimal Curves in Homogeneous Reductive Spaces of the Unitary Group of a Finite von Neumann Algebra

We study the metric geometry of homogeneous reductive spaces of the unitary group of a finite von Neumann algebra with a non complete Riemannian metric. The main result gives an abstract sufficient condition in order that the geodesics of the Levi-Civita connection are locally minimal. Then, we show how this result applies to several examples. To my family     

The Gellerstedt problem for a parabolic-hyperbolic equation of the second kind

In this paper we consider the Gellerstedt problem for a parabolic-hyperbolic equation of the second kind. We prove the unique solvability of this problem by means of a new representation for a solution to the modified Cauchy problem in a generalized class R.     

Rational approximation to Neumann series of Bessel functions

LetJ _n (z) be the Bessel function of the first kind and ordern, and letf(z) be an analytic function in|z|r (r>0); then it is known that the Bessel expansion

The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations

We investigate the large-time behavior of viscosity solutions of the Cauchy-Dirichlet problem (CD) for Hamilton-Jacobi equations on bounded domains. We establish general convergence results for viscosity solutions of (CD) by using the Aubry-Mather theory.      

The parity problem of polymatroids without double circuits

According to the present state of the theory of the matroid parity problem, the existence of a good characterization to the size of a maximum matching depends on the behavior of certain substructures, called double circuits. In this paper we prove that if a polymatroid has no double circuits then a partition type min-max formula characterizes the size of a maximum matching. Applications to parity constrained orientations and to a rigidity problem are given. Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No. 504438.     

Boundary regularity for the\bar \partial _b - Neumann problem, part 1problem, part 1

We establish sharp regularity and Fredholm theorems for the operator on domains satisfying some nongeneric geometric conditions. We use these domains to construct explicit examples of bad behavior of the Kohn Laplacian: It is not always hypoelliptic up to the boundary, its partial inverse is not compact and it is not globally subelliptic.