Gellerstedt problem with nonclassical matching conditions for the solution gradient on the type change line with data on internal characteristics

We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation. An initial function is posed in the ellipticity domain of the equation on the boundary of the unit half-circle with center the origin. Zero conditions are posed on characteristics in the hyperbolicity domain of the equation. “Frankl-type conditions” are posed on the type change line of the equation. We show that the problem is either conditionally solvable or uniquely solvable. We obtain a closed-form solvability condition in the case of conditional solvability. We derive integral representations of the solution in all cases.     

On solvability of boundary integral equations of potential theory for a multidimensional cusp domain

The Dirichlet and the Neumann problems for the Laplace equation on a multidimensional cusp domain are considered. The unique solvability of the boundary integral equation for the internal Dirichlet problem for harmonic double layer potential is established. We also prove the unique solvability of the boundary integral equation for the external Neumann problem for harmonic single layer potential. Bibliography: 13 titles.     

On the blow-up of the solution of an equation with a gradient nonlinearity

We continue the study of a nonlinear third-order equation of the Hamilton-Jacobi type. For this equation, we consider an initial-boundary value problem in a bounded domain with smooth boundary and prove the local solvability in the strong generalized sense; in addition, we derive sufficient conditions for the blow-up in finite time and sufficient conditions for the time-global solvability.     

On the blow-up of the solution of an equation related to the Hamilton-Jacobi equation

A new model three-dimensional third-order equation of Hamilton-Jacobi type is derived. For this equation, the initial boundary-value problem in a bounded domain with smooth boundary is studied and local solvability in the strong generalized sense is proved; in addition, sufficient conditions for the blow-up in finite time and sufficient conditions for global (in time) solvability are obtained.     

Problem with generalized fractional differentiation operators for a hyperbolic equation degenerating in the interior of the domain

We study the unique solvability of a nonlocal problem with generalized fractional differentiation operators in the boundary condition for a hyperbolic equation degenerating in the interior of the domain.     

On the solution of the Gellerstedt problem for the Lavrent’ev-Bitsadze equation

We study the solvability of the Gellerstedt problem for the Lavrent??ev-Bitsadze equation under an inhomogeneous boundary condition on the half-circle of the ellipticity domain of the equation, homogeneous boundary conditions on external, internal, and parallel side characteristics of the hyperbolicity domain of the equation, and the transmission conditions on the type change line of the equation.     

Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.     

Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.     

On a boundary value problem with shift for an equation of mixed type in an unbounded domain

We study the unique solvability of a problem with shift for an equation of mixed type in an unbounded domain. We prove the uniqueness theorem under inequality-type constraints for known functions for various orders of the fractional differentiation operators in the boundary condition. The existence of a solution is proved by reduction to a Fredholm equation of the second kind, whose unconditional solvability follows from the uniqueness of the solution of the problem.     

Generalized boundary values of the solutions of semilinear elliptic equations from weighted functional spaces

In weighted C-spaces, we establish the solvability of a boundary-value problem for a semilinear elliptic equation of order 2m in a bounded domain with generalized functions given on its boundary, strong power singularities at some points of the boundary, and finite orders of singularities on the entire boundary. The behavior of the solution near the boundary of the domain is analyzed.     

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.     

On an inhomogeneous problem for parabolic-hyperbolic equation

We study a boundary value problem for an inhomogeneous parabolic-hyperbolic equation with a noncharacteristic type change line. Boundary conditions of the first kind are posed on characteristics in the parabolic and hyperbolic parts of the domain where the equation is given, and a condition of the third kind is posed on the noncharacteristic part of the boundary in the parabolic part. First, we study the solvability of an inhomogeneous initial–boundary value problem for a parabolic equation.     

一阶非线性椭圆型复方程的间断Riemam边值问题

§1 问题的提法 本文讨论一阶非线性一致椭圆型复方程 的间断非线性Riemann边值问题。以E表示复平面,D~ 表示有界N 1连通区域,其边界由     

Euler-Poisson-Darboux differential-operator equation with variable domains of smooth operators

We prove the strong well-posed solvability of the Cauchy problem for a second-order singular hyperbolic differential equation with variable domain of variable unbounded operator coefficients and for the mixed problem for a complete equation of string vibrations with a strong singularity in time and with a time-dependent boundary condition.     

Construction and Structure Properties of Solutions of a Periodic Boundary Value Problem for a Generalization of the Matrix Lyapunov and Riccati Equations

We obtain constructive sufficient conditions for the unique solvability of a periodic boundary value problem for a matrix differential equation that generalizes the Lyapunov and Riccati equations, develop an algorithm for constructing the solution of this equation, estimate the domain where the solution is localized, and study the structural properties of the solution.     

An over-determined boundary problem for the Helmholtz equation in a semiinfinite domain with a curvilinear boundary

In this paper we consider an over-determined Cauchy problem for the Helmholtz equation in a semiinfinite domain with a piecewise smooth curvilinear boundary. Applying the Fourier transform method in the space of distributions of slow growth, we establish the necessary and sufficient solvability conditions which connect the boundary functions. We construct integral representations of a solution.     

ON THE PLANE STRESS BOUNDARY VALUE PROBLEM OF QUASI-STATIC LINEAR THERMOELASTICITY

1.IntroductionTheinvestigationoftheinfluenceofthetemperaturedistributioninanelasticbodyontheinternalstressesandstrainsisamainillterestofthetheoryofthermoelasticity.Typically,theinitialtemperaturedistributionandtheboundaryvalueoftemperatureinanintervaloftimeaswellasacertaindisplacementorstressorcompoundboundaryvalueconditionsaregiven,alltheaboveareutilizedsoastodeterminethestresses,strainsandthetemperaturedistributionintheelasticbody,inanintervaloftime.Werestrictouxattentiontoelasticequilibrium…     

On boundary-value problems for a second-order differential equation with complex coefficients in a plane domain

We study boundary-value problems for a homogeneous partial differential equation of the second order with arbitrary constant complex coefficients and a homogeneous symbol in a bounded domain with smooth boundary. Necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. These conditions are written in the form of a moment problem on the boundary of the domain and applied to the investigation of boundary-value problems. This moment problem is solved in the case of a disk.     

The dirichlet problem for a loaded mixed-type equation in a rectangular domain

We establish necessary and sufficient conditions for the unique solvability of the first boundary problem for a loaded equation with the Lavrent’ev-Bitsadze operator in a rectangular domain. We obtain a solution to the stated problem as the sum of the eigenfunction series for the corresponding one-dimensional problem with respect to eigenvalues. We prove the stability of the solution with respect to boundary functions.     

Neumann and Robin problems in a cracked domain with jump conditions on cracks

The boundary value problem for the Laplace equation is studied on a domain with smooth compact boundary and with smooth internal cracks. The Neumann or the Robin condition is given on the boundary of the domain. The jump of the function and the jump of its normal derivative is prescribed on the cracks. The solution is looked for in the form of the sum of a single layer potential and a double layer potential. The solvability of the corresponding integral equation is determined and the explicit solution of this equation is given in the form of the Neumann series. Estimates for the absolute value of the solution of the boundary value problem and for the absolute value of the gradient of the solution are presented.