On the solvability problem for equations with a single coefficient

It is proved that the general solvability problem for equations in a free group is polynomially reducible to the solvability problem for equations of the formw(x ₁, ...,x _n)=g, whereg is a coefficient, i.e., an element of the group, andw(x ₁, ...,x _n) is a group word in the alphabet of unknowns. We prove the NP-completeness of the solvability problem in a free semigroup for equations of the formw(x ₁,...,x _n)=g, wherew(x ₁,...,x _n) is a semigroup word in the alphabet of unknowns andg is an element of a free semigroup. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 832–845, June, 1996. I wish to express my deep gratitude to S. I. Adyan and A. A. Razborov for the discussion of the present paper and for valuable remarks concerning the exposition. This research was partially supported by the International Science Foundation under grant MUV000.     

On solvability of a two-sided singular control problem

We study a two-sided singular control problem in a general linear diffusion setting and provide a set of conditions under which an optimal control exists uniquely and is of singular control type. Moreover, under these conditions the associated value function can be written in a quasi-explicit form. Furthermore, we investigate comparative static properties of the solution with respect to the volatility and control parameters. Lastly we illustrate the results with two explicit examples.     

On solvability of the Cauchy problem for one quasilinear singular functional-differential equation

We consider the Cauchy problem with zero initial conditions for quasilinear singular functional-differential equation of the second order with a delay at singular summand. We obtain sufficient conditions of solvability of the problem.     

On solvability of singular integral-differential equations with convolution

In this paper, we study a class of singular integral-different equations of convolution type with Cauchy kernel. By means of the classical boundary value theory, of the theory of Fourier analysis, and of the principle of analytic continuation, we transform the equations into the Riemann-Hilbert problems with discontinuous coefficients and obtain the general solutions and conditions of solvability in class $\{0\}$. Thus, the result in this paper generalizes the classical theory of integral equations and boundary value problems.     

On solvability of the Dirichlet problem to the semilinear Schrödinger equation with singular potential

We prove the existence of a positive solution of the Dirichlet problem for the Schrödinger equation whose potential possesses a critical singularity. It is assumed that the boundary of the domain is average concave in a neighborhood of the origin. Bibliography: 13 titles.     

On the solvability of nonlinear differential inequalities with singular coefficients

On the basis of the nonlinear capacity method, we obtain exact solvability conditions for some classes of stationary and evolution nonlinear differential inequalities with singular coefficients.     

On solvability of the jump problem

Let Γ be a closed non-rectifiable Jordan curve on the complex plane C. We consider the so-called jump problem, i.e. the boundary value problem for determination of a holomorphic in function with a given jump on Γ. The main result is a condition of solvability of the problem in terms of a new metric dimension of the curve.     

On Riemann curvature of singular square metrics

Square metrics is an important class of Finsler metrics. Recently, we introduced a special class of non-regular Finsler metrics called singular square metrics. The main purpose of this paper is to provide a necessary and sufficient condition for singular square metrics to be of constant Ricci or flag curvature when dimension $n\ge 3$.     

On the solvability of a singular integral equation with a non-Carleman shift

We consider a singular integral equation with a non-Carleman shift on an interval. We prove the unique solvability of this equation in weighted Hölder classes under certain restrictions on the coefficients. We show that the solution of the equation can be written in quadratures.     

On the existence of diffusions with singular drift coefficient

Let L:be an operator in R~d,where the matrix(a) is bounded,Holder continuous and uniformly positive definite,and (b~t(x)) is Borel measurable.In this paper we prove the existence of L-diffusion under the hypothesis that where g_1(z)=1,g_2(z)=-ln|z|and g_c(z)=|z|~(2-d) for d≥3.     

On solvability of three spectra problem

Three spectral problems generated by the same Sturm–Liouville equation are considered: Neumann–Dirichlet problem (the Neumann condition at the left end and the Dirichlet condition at the right end) on the whole interval [0, a], Neumann–Dirichlet problem on and Dirichlet–Dirichlet problem on . The three spectra inverse problem, i.e. the problem of recovering the Sturm–Liuville equation using the three spectra of these boundary value problems is completely solved.     

On the weak solvability of the problem of viscoelasticity with memory

The existence of a weak solution is proved for a certain Oldroyd model of motion of a viscoelastic medium that allows for the memory of the system. The proof uses the theory of regular Lagrange flows and a topological approximation method that reduces the posed problem to an operator equation, its ε-regularization in smoother spaces, the use of a priori estimates and a topological degree for the proof of the solvability of the ε-regularized equations, and the passage to the limit as ε → 0.     

On the solvability of a nonlinear pseudoparabolic problem

This paper is concerned with the study of an initial boundary value problem for a nonlinear second order pseudoparabolic equation arising in the unidirectional flow of a thermodynamic compatible third grade fluid. We establish some a priori bounds for the solution and prove its existence.     

On a boundary-value problem solvability condition

We prove a sufficient condition for the solvability of a two-point problem.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 247–250, February, 1973.