Best-possible estimates are obtained for solutions of the Dirichlet problem for linear elliptic nondivergence equations of second order near a conical point of the boundary of the domain under minimal conditions on the smoothness of the coefficients of the equation.
The Navier-Stokes system is considered in a plane domain that has several exits to infinity having the form of channels of bounded width. It is assumed that the external force decays sufficiently fast at infinity. Solutions are considered that are defined and bounded for all t is-an-element-of R. Such solutions lie on an attractor of the system. An asymptotic expansion as Absolute value of x --> infinity is obtained for these solutions. The presence of this expansion indicates, in particular, that turbulence in this situation does not propagate to infinity.
Semicontinuous real functions are considered. The following property is established for the Dini directional semiderivative and the Dini semidifferential (the subdifferential). If at some point the semiderivative is positive in a convex cone of directions, then arbitrarily close to the point under consideration there exists a point at which the function is subdifferentiable and has a subgradient belonging to the positively dual cone. This result is used in the theory of the Hamilton-Jacobi equations to prove the equivalence of various types of definitions of generalized solutions.
A special commutative Moufang loop H of order 3(4) is described. With the help of this loop, a trilinear Dickson form is constructed whose automorphism group is a Chevalley group of type E6. Next, with the help of H, a 27-dimensional representation is constructed for O7(3) over Z[zeta] , zeta3 = 1 . This makes it possible to prove anew the embedding O7 (3) subset-or-equal-to 2 E6(2). A similar construction concerning the embedding L4(3) subset-or-equal-to F4(2) is described.
A description is given of situations in which a subspace invariant with respect to a differential operator with constant coefficients admits spectral synthesis. Bibliography: 22 titles.
