Asymptotic Behavior of Solutions of Inverse Problems for Degenerate Parabolic Equations

We obtain theorems on the proximity as t → +∞ between the solution of the inverse problem for a second-order degenerate parabolic equation with one spatial variable and the solution of the inverse problem for a second-order degenerate ordinary differential equation under an additional integral observation condition. The conditions imposed on the input data admit oscillations of the functions on the right-hand side in the parabolic equation under study.     

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.     

Richardson extrapolation in boundary value problems for differential equations with nonregular right-hand side

When the finite-difference method is used to solve initial- or boundary value problems with smooth data functions, the accuracy of the numerical results may be considerably improved by acceleration techniques like Richardson extrapolation. However, the success of such a technique is doubtful in cases were the right-hand side or the coefficients of the equation are not sufficiently smooth, because the validity of an asymptotic error expansion — which is the theoretical prerequisite for the convergence analysis of the Richardson extrapolation — is not a priori obvious. In this work we show that the Richardson extrapolation may be successfully applied to the finite-difference solutions of boundary value problems for ordinary second-order linear differential equations with a nonregular right-hand side. We present some numerical results confirming our conclusions.     

Solvability of discrete Neumann boundary value problems

In this article we gain solvability to a nonlinear, second-order difference equation with discrete Neumann boundary conditions. Our methods involve new inequalities on the right-hand side of the difference equation and Schaefer's Theorem in the finite-dimensional space setting.     

Singular Dirichlet second-order BVPs with impulses

The paper deals with the existence of solutions to singular second-order differential equations with impulse effects and with the Dirichlet boundary conditions. The right-hand side of the differential equation can be singular in its phase variable.     

二阶微分包含的边值问题及其应用

在Banach中讨论了一类二阶微分包含的边值问题和一类反馈控制问题.结合不动点定理,对解的存在性给出了两个充分性条件,最后应用前面的结果建立了反馈控制问题解的存在性定理.     

Differentiation of the functional in a parametric optimization problem for a coefficient of a semilinear elliptic equation

We study parametric optimization with respect to an integral criterion of the higher coefficient and the right-hand side of a second-order semilinear elliptic equation with the Dirichlet boundary condition. We obtain formulas for the first partial derivatives of the objective functional with respect to the control parameters. The total preservation (preservation for the entire set of control parameters) of the unique solvability of the boundary value problem for this equation is proved based on the theory of monotone operators.     

A Delay Second-Order Set-Valued Differential Equation with Hukuhara Derivatives

In this article, we study a second-order differential equation with three-point boundary conditions with the notion of Hukuhara derivatives. The existence and uniqueness of a solution is given under a Lipschitz condition on the right-hand side in the second and third variables.     

Dirichlet problem for a third-order equation of mixed type in a rectangular domain

For a third-order differential equation of parabolic-hyperbolic type, we suggest a method for studying the first boundary value problem by solving an inverse problem for a second-order equation of mixed type with unknown right-hand side. We obtain a uniqueness criterion for the solution of the inverse problem. The solution of the inverse problem and the Dirichlet problem for the original equation is constructed in the form of the sum of a Fourier series.     

Solution of the Bitsadze–Samarskii problem for a parabolic system in a semibounded domain

We consider a nonlocal initial–boundary value Bitsadze–Samarskii problem for a spatially one-dimensional parabolic second-order system in a semibounded domain with nonsmooth lateral boundary. The boundary integral equation method is used to construct a classical solution of this problem under the condition that the vector function on the right-hand side in the nonlocal boundary condition only has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

On boundary value problems for an <Emphasis Type="Italic">n</Emphasis>th-order equation

For an nth-order differential equation with right-hand side satisfying the Carathéodory conditions, we consider a two-point boundary value problem with boundary conditions of which the first two have a general form and the remaining conditions are simplest. We show that the existence of a lower and an upper function and the compactness condition imply the solvability of this boundary value problem.     

Periodic solutions for second order differential inclusions with the scalar p-Laplacian

We study periodic problems driven by the scalar p-Laplacian with a multivalued right-hand side nonlinearity. We prove two existence theorems. In the first, we assume nonuniform nonresonance conditions between two successive eigenvalues of the negative p-Laplacian with periodic boundary conditions. In the second, we employ certain Landesman-Lazer type conditions. Our approach is based on degree theory.     

Extremal solutions of boundary value problems

To prove the existence of a solution of a two-point boundary value problem for an nth-order operator equation by the a priori estimate method, we study extremal solutions of auxiliary boundary value problems for an nth-order differential equation with simplest right-hand side, which have a unique solution under certain restrictions on the boundary conditions.     

Regularity of solutions of boundary value problems for a second-order parabolic equation in weighted Hölder spaces

We consider the first boundary value problem and the oblique derivative problem for a linear second-order parabolic equation in noncylindrical not necessarily bounded domains with nonsmooth (with respect to t) and noncompact lateral boundary under the assumption that the right-hand side and the lower-order coefficients of the equation may have certain growth when approaching the parabolic boundary of the domain and all coefficients are locally Hölder with given characteristics of the Hölder property. We construct a smoothness scale of solutions of these boundary value problems in Hölder spaces of functions that admit growth of higher derivatives near the parabolic boundary of the domain.     

Inverse problem of simultaneously determining the right-hand side and the coefficient of a lower order derivative for a parabolic equation on the plane

We obtain existence and uniqueness theorems for the solution of the inverse problem of simultaneously determining the right-hand side and the coefficient of a lower-order derivative in a parabolic equation under an integral observation condition. We give explicit estimates for the maximum absolute value of the unknown right-hand side and the unknown coefficient of the equation with constants expressed via the input data of the problem. We present a nontrivial example of an inverse problem to which our theorems apply.     

Coefficient inverse problem for Poisson’s equation in a cylinder

The inverse problem of determining the coefficient on the right-hand side of Poisson’s equation in a cylindrical domain is considered. The Dirichlet boundary value problem is studied. Two types of additional information (overdetermination) can be specified: (i) the trace of the solution to the boundary value problem on a manifold of lower dimension inside the domain and (ii) the normal derivative on a portion of the boundary. (Global) existence and uniqueness theorems are proved for the problems. The study is performed in the class of continuous functions whose derivatives satisfy a Hölder condition.     

How to Approach Nonstandard Boundary Value Problems

A new approach to nonstandard boundary value problems is suggested. For such problems, we construct equivalent inclusions with surjective operators and study the solvability of these inclusions. The paper consists of two parts. The first part deals with problems in which the right-hand side of the equation is a Lipschitz mapping (Section 3); in the second part (Section 4), this mapping is completely continuous with respect to a surjective operator A. The paper also gives examples of how our theorems can be applied when studying nonstandard boundary value problems.     

On some boundary value problems for a system of two second-order equations

For a two-point homogeneous boundary value problem for a system of two nonlinear second-order differential equations, we suggest sufficient solvability conditions (in particular, stated, like Bernstein conditions, in terms of the growth of the absolute values of the right-hand sides of the system with respect to the derivatives of the unknown functions). We obtain a priori estimates for solutions.     

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

Inverse problems for elliptic equations in the space: II

We consider inverse problems of finding the right-hand side of a linear second-order elliptic equation of general form. The first boundary value problem is studied. We consider two ways of indicating additional information (overdetermination): the trace of the solution can be given on some lower-dimensional manifold inside the domain, or the normal derivative can be specified on part of the boundary. On the basis of the Fredholm alternative proved in the first part of the present paper for the inverse problems in question, we single out conditions on the given functions under which the inverse problem is uniquely solvable. Various types of such conditions are considered. The study is carried out in the class of continuous functions whose derivatives satisfy the Hölder condition.