Solvability conditions of a boundary value problem with operator coefficients and related estimates of the norms of intermediate derivative operators

Sufficient conditions for the proper and unique solvability in the Sobolev space of vector functions of the boundary value problem for a certain class of second-order elliptic operator differential equations on a semiaxis are obtained. The boundary condition at zero involves an abstract linear operator. The solvability conditions are established by using properties of operator coefficients. The norms of intermediate derivative operators, which are closely related to the solvability conditions, are estimated.     

Existence and blow-up of solutions of a pseudoparabolic equation

We obtain sufficient blow-up conditions for the solution of a nonlinear differential problem with given initial and boundary conditions. We prove the solvability of this problem in any finite cylinder under some restrictions on the nonlinear operators.     

Controlling linear functional differential equation

Necessary conditions for the optimal control of a linear system of neutral type functional differential equations are obtained. We show that a success in reducing the initial problem to the problem of solvability of a certain boundary value problem is, in principle, a question of one's ability to construct adjoint operators to the operators appearing in the initial control problem.     

Solvability of a nonlocal problem for a loaded parabolic-hyperbolic equation

For a mixed-type equation we study a problem with generalized fractional integrodifferentiation operators in the boundary condition. We prove its unique solvability under inequality-type conditions imposed on the known functions for various orders of fractional integrodifferentiation operators. We prove the existence of a solution to the problem by reducing the latter to a fractional differential equation.     

Problem without Initial Conditions for a Class of Inverse Parabolic Operator-Differential Equations of Third Order

In a weighted Sobolev-type space, the well-posedness and unique solvability of a problem without initial conditions for a third-order operator-differential equation with an inverse parabolic principal part are established. The solvability conditions are expressed in terms of the operator coefficients of the equation. Additionally, the norms of the operators of intermediate derivatives closely related to the solvability conditions are estimated. The relation between the weight exponent and the lower boundary of the spectrum of the basic operator involved in the principal part of the equation is established. Note that the principal part of the equation has a multiple characteristic.     

On the solvability in a weighted space of an initial–boundary value problem for a third-order operator-differential equation with a parabolic principal part

In a Sobolev-type space with an exponential weight, sufficient conditions are obtained for the correct and unique solvability of an initial–boundary value problem for a third-order operator-differential equation with a parabolic principal part having a multiple characteristic. The conditions are expressed in terms of the operator coefficients of the equation. Additionally, the norms of the operators of intermediate derivatives associated with the solvability conditions are estimated. The relation between the weight exponent and the lower boundary of the spectrum of the basic operator involved in the principal part of the equation is established. The results are illustrated as applied to a mixed problem for partial differential equations.     

Multipoint problem for a class of evolution equations

We prove the well-posed solvability of a nonlocal problem for evolution equations with nonnegative self-adjoint operators that have discrete spectrum and with boundary conditions in the space of linear continuous functionals of the type of ultra-distributions identified with formal Fourier series.     

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.     

The inverse problem for differential operators of second order with regular boundary conditions

This paper is devoted to the proof of the unique solvability of the inverse problem for second-order differential operators with arbitrary regular nonseparable boundary conditions. It is shown that the operator can be recovered from three of its spectra. As a special case, the well-known reconstruction of the Sturm-Liouville operator is accomplished.     

Solvability conditions for infinite systems of difference equations

The Fredholm property of some linear infinite dimensional difference operators is studied. In the case corresponding to discretization of differential equations on the real axis, the index of the corresponding operators is computed and solvability conditions for the nonhomogeneous problem are established. In the multi-dimensional case, conditions of the normal solvability of the corresponding discrete operators are formulated in terms of limiting problems. The results on the location of the spectrum and the solvability conditions allow various applications to linear and nonlinear problems.     

To the theory of boundary-value problems for elliptic equations with superposition operators in the boundary condition. I

We study the existence, uniqueness, and constant sign property of classical solutions to a nonlocal boundary-value problem for a second-order elliptic equation in a bounded domain of the Euclidean space. Using the system of maps that define superposition operators, we construct some subset of the domain boundary and establish the connection between the solvability of the problem under consideration and the solvability of the boundary value equation on the constructed subset.     

A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type

We consider a mixed-type equation whose order degenerates along the line of change of type. For this equation we study the unique solvability of a nonlocal problem with the Saigo operators in the boundary condition. We prove the uniqueness theorem under certain conditions (stated as inequalities) on known functions. To prove the existence of solution to the problem, we equivalently reduce it to a singular integral equation with the Cauchy kernel. We establish a condition ensuring the existence of a regularizer which reduces the obtained equation to a Fredholm equation of the second kind, whose unique solvability follows from that of the problem.     

On Solvability of Nonlocal Problem for Loaded Parabolic-Hyperbolic Equation

We study unique solvability of a nonlocal problem for equations of mixed type in a finite domain. This equation contains the partial fractional Riemann–Liouville derivative. The boundary condition of the problem contains a linear combination of operators of fractional differentiation in the sense of Riemann–Liouville of values of function derivative on the degeneration line and generalized operators of fractional integro-differentiation in the sense of M. Saigo. The uniqueness theorem of the problem is proved by a modified Tricomi method. The existence of solutions is equivalently reduced to the solvability of Fredholm integral equation of the second kind.     

Strong Solvability of Boundary Value Contact Problems

Strong solvability in Sobolev spaces is proved for a unilateral contact boundary value problem for a class of nonlinear discontinuous operators. The operator is assumed to be of Caratheodory type and to satisfy a suitable ellipticity condition. Only measurability with respect to the independent variable x is required. The main tool of the proof is an estimate for the second derivatives of the functions which satisfy the unilateral boundary conditions, in which it has been possible to prove that the constant is equal to 1.     

Riemann–Hilbert-type Boundary Value Problems on a Half Hexagon

In this article we discuss the explicit solvability of both Schwarz boundary value problem and Riemann–Hilbert boundary value problem on a half hexagon in the complex plane. Schwarz-type and Pompeiu-type integrals are obtained. The boundary behavior of these operators is discussed.Finally, we investigate the Schwarz problem and the Riemann–Hilbert problem for inhomogeneous Cauchy–Riemann equations.     

The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space

We consider linear fractional differential operator equations involving the Caputo derivative. The goal of this paper is to establish conditions for the unique solvability of the inverse Cauchy problem for these equations. We use properties of the Mittag-Leffler function and the calculus of sectorial operators in a Banach space. For equations with operators in a general form we obtain sufficient conditions for the unique solvability, and for equations with densely defined sectorial operators we obtain necessary and sufficient unique solvability conditions.     

The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients

The present paper discusses relations between regularity, Dirichlet, and Neumann problems. We investigate the boundary problems for block operators and prove, in particular, that the solvability of the regularity problem does not imply the solvability of the dual Dirichlet problem for general elliptic operators with complex bounded measurable coefficients. This is strikingly different from the case of real operators, for which such an implication was established in 1993 by C. Kenig, J. Pipher [Invent. Math. 113 (3) (1993) 447-509] and since then has served as an integral part of many results.     

Solvability of some boundary-value problems for polyharmonic equation with Hadamard-Marchaud boundary operator

We investigate solvability conditions for certain non-classical boundary-value problems for the polyharmonic equation. As the boundary operators we consider fractional differential operators in the sense of Hadamard-Marchaud. The considered problems generalize well-known Dirichlet and von Neumann boundary-value problems for boundary operators of fractional type.     

On estimates for solutions of elliptic boundary value problems in a layer

We consider boundary value problems for elliptic operators with constant coefficients in a layer, i.e., in a domain between two parallel planes. We assume that the Lopatinskii condition and the condition of the unique solvability of an auxiliary problem for an ordinary differential operator are satisfied. We prove theorems on the solvability and smoothness of solutions in Sobolev spaces with weight of exponential type.     

On the solvability of the dirichlet problem for elliptic nondivergent equations of the second order in a domain with conical point

We study the problem of solvability of the Dirichlet problem for second-order linear and quasilinear uniformly elliptic equations in a bounded domain whose boundary contains a conical point. We prove new theorems on the unique solvability of a linear problem under minimal smoothness conditions for the coefficients, right-hand sides, and the boundary of the domain. We find classes of solvability of the problem for quasilinear equations under natural conditions.