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 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 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.     

On the solvability of an abstract Cauchy problem

We consider the Cauchy problem for a first-order linear inhomogeneous differential equation for functions ranging in a Banach space in the case of a sectorial operator coefficient. We find conditions for the solvability of the Cauchy problem for the case in which the right-hand side is not necessarily a Hölder function.     

On solvability conditions of the Dirichlet problem form-curvature equations

This papor contains a refined variant of the Jenkins-Serrin-Trudinger condition related to the Dirichlet problem for curvature equations. The new condition implies the classical solvability of the Gauss curvature equation in the case of positive curvature small enough on the boundary. Bibliography: 8 titles. Dedicated to V. A. Solonnikov on his sixtieth anniversary Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 48–65. Translated by S. A. Titarenko.     

On the solvability of the problem of saturated-unsaturated filtration consolidation

We prove the existence of a solution of the initial-boundary value problem modeling the process of liquid filtration in a viscoelastic medium in the case of semipermeability on part of the boundary. To determine the generalized solution, we use the Kirchhoff transform. We consider the case (most often used in applications) in which the range of the Kirchhoff function is only part of the real line. To prove the existence theorem, we use the method of the semidiscretization with respect to the variable t and the Galerkin method.     

The hilbert boundary value problem for generalized analytic functions in clifford analysis

Let ?_0,n be the real Clifford algebra generated by e₁, e₂,…, e_n satisfying e_ie_j+e_je_i=−2δ_ij, i, j=1,2,…, n. e₀ is the unit element. Let Ω be an open set. A function f is called left generalized analytic in Ω if f satisfies the equation

equation(0.1)

Lf=0,$Lf=0,$

where

L=q₀e₀∂x₀+ q₁e₁∂x₁+…+qnen_∂xn,$L=q_0{e}_0\partial x_0+\hspace{0.17em}{q}_1{e}_1\partial x_1+\dots +q_n{e}_n{\partial }_{x_n},$

qi <0, i=0,1,…, n. In this article, we first give the kernel function for the generalized analytic function. Further, the Hilbert boundary value problem for generalized analytic functions in ?ⁿ⁺¹₊ will be investigated.     

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 the solvability of a three-dimensional problem of thermocapillary convection

The solvability of a stationary problem with a free boundary, for a system of Navier-Stokes equations and an equation of thermal balance of a viscous incompressible fluid that partially fills a vessel, is established. It is assumed that the surface tension coefficient of the fluid depends on the temperature and that the difference between the temperature of the wall of the vessel and of its surroundings is nonzero but sufficiently small.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 18–27, 1990.     

On the solvability of a self-similar problem of forced convection

We consider a problem, described by a third order ordinary differential equation, that concerns nonisothermal flow of a viscous fluid through pipes. A study is made of a nonlinear boundary value problem for specified values of the thermal flow and a given Peclet number. Non-solvability of the boundary value problem is shown for parameter values larger than some critical value.Translated from Dinamicheskie Sistemy, No. 5, pp. 59–62, 1986.     

On normal solvability of the Riemann problem with singular coefficient

Suppose is a singular matrix function on a simple, closed, rectifiable contour . We present a necessary and sufficient condition for normal solvability of the Riemann problem with coefficient in the case where admits a spectral (or generalized Wiener-Hopf) factorization with essentially bounded. The boundedness of is not required when takes injective values a.e. on .

     

On the solvability of the higher order elliptic boundary value problem

In this paper, by employing a non-variational version of a max–min principle, new results on existence and uniqueness of generalized solution to the boundary value problem of higher order elliptic partial differential equations at resonance are given.     

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 sufficient conditions for the positive solvability of the Riemann-Hilbert problem

Translated from Matematicheskie Zametki, Vol. 51, No. 2, pp. 9–19, February, 1992.     

On solvability of the Dirichlet problem for the third-order hyperbolic equation

We consider a Dirichlet problem for the third-order hyperbolic equation and show the existence and uniqueness of its classical solution. For the proof of unique solvability, we use the methods of Riemann’s function and integral equations.