Integral operators determined by quasielliptic equations. I

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 6, pp. 52–67, November–December, 1993.     

Spectrum of quasielliptic differential operators

Translated from Matematicheskie Zametiki, Vol. 53, No. 6, pp. 145–148, June, 1993.     

Implicit functions determined by equations with unbounded operators and the solvability of operator equations

For implicit functions determined by the equation T_x+F(λ,x)=0, where T is an unbounded operator, existence and differentiability conditions are established. As an application, the solvability conditions for equations of the form Tx+f(x)=0 are derived. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 216–224. Translated by L. Yu. Kolotilina.     

Quasielliptic operators and Sobolev type equations. II

We consider matrix quasielliptic operators on the whole space. Under the quasihomogeneity condition for symbols, we establish the isomorphism theorem for these operators in the special scales of Sobolev spaces. In particular, this result implies a series of available isomorphism theorems for elliptic operators and theorems about the unique solvability of the initial value problem for a broad class of systems of Sobolev type.     

Integral equations of the third kind with unbounded operators

We consider linear functional equations of the third kind in L ₂ with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.     

Relativistic equations determined by the operators of the exterior derivative and generalized divergence

Theoretical and Mathematical Physics -     

On removable singularities of bounded solutions to quasielliptic equations

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No 4, pp. 3–14, July–August, 1992.     

Integral operators for elliptic equations in three independent variables,ii

An integral operator is obtained which maps analytic functions of two complex variables onto solutions of a homogeneous linear elliptic partial differential equation in three independent variables. An inversion formula is given and used to construct a complete family of solutions for the elliptic equation under investigation     

Integral operators generated by exponential integrals

Some results for integral operators whose kernels are the exponential integrals are presented. In particular, their interrelations and commutativity properties with the differentiation operator, and the behavior of the image functions near the endpoints of the domain of integration, are investigated.     

Integral operators generated by H-continuous measures

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 6, pp. 769–773, June, 1989.     

The membership of solutions of quasielliptic equations to space LP

It is established that the solutions of a quasielliptic equation, belonging to space L₁ with weight equal to a negative power of the distance to the flat part of the boundary, belong to space L_P with some p>1. In particular, the positive solutions of uniformly elliptic equations in bounded regions with a smooth boundary belong to L_p() with any p相似文献   

Weak solutions for equations defined by accretive operators II: relaxation limits

In the first part of this paper we define solutions for certain nonlinear equations defined by accretive operators, “dissipative solution”. This kind of solution is equivalent to the viscosity solutions for Hamilton-Jacobi equations and to the entropy solutions for conservation laws.In this paper we use dissipative solutions to obtain several relaxation limits for systems of semilinear transport equations and quasilinear conservation laws. These converge to diffusion second-order equations and in one case to a single conservation law. The relaxation limit is obtained using a version of the perturbed test function method to pass to the limit. This guarantees existence for the considered equations.     

Differential properties of solutions of equations which are quasielliptic at infinity

Siberian Mathematical Journal -     

Lebesgue-Stieltjes形式的Choquet积分Ⅱ

研究了Lebesgue-Stieltjes形式的Choquet积分的收敛性定理,如单调收敛定理、法都引理、控制收敛定理等.