Nonlocal Boundary Value Problems for Elliptic-Parabolic Differential and Difference Equations

Abstract

The abstract nonlocal boundary value problem for differential equations in a Hilbert space with the self-adjoint positive definite operator is considered. The well-posedness of this problem in Hölder spaces with a weight is established. The coercivity inequalities for the solution of boundary value problems for elliptic-parabolic equations are obtained. The first order of accuracy difference scheme for the approximate solution of this nonlocal boundary value problem is presented. The well-posedness of this difference scheme in Hölder spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained.

1. Introduction

It is known that various problems in fluid mechanics and other areas of engineering, physics, and biological systems lead to partial differential equations of variable types. Methods of solutions of nonlocal boundary value problems for partial differential equations of variable type have been studied extensively by many researchers (see, e.g., [1–4] and the references given therein).

The nonlocal boundary value problemfor differential equations in a Hilbert space with the self-adjoint positive definite operator is considered.

Let us denote by the Banach space obtained by completion of the set of all smooth -valued function on in the normand denote by the Banach space obtained by completion of the set of all smooth -valued function on in the normfinally denote by the Banach space obtained by completion of the set of all smooth -valued function on in the normHere stands for the Banach space of all continuous functions defined on with values in equipped with the normA function is called a solution of problem (1.1) if the following conditions are satisfied.

(i) is twice continuously differentiable on the segment and continuously differentiable on the segment ; the derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii)The element belongs to the domain of for all , and the function is continuous on the segment .(iii) satisfies the equations and the nonlocal boundary condition (1.1).

A solution of problem (1.1) defined in this manner will henceforth be referred to as a solution of problem (1.1) in the space .

We say that problem (1.1) is well-posed in if there exists a unique solution in of problem (1.1) for any and , and the following coercivity inequality is satisfied:where is independent of and

Problem (1.1) is not well-posed in () [5]. The well-posedness of the boundary value problem (1.1) can be established if one considers this problem in certain spaces of smooth -valued functions on .

A function is said to be a solution of problem (1.1) in if it is a solution of this problem in and the functions and belong to .

As in the case of the space we say that problem (1.1) is well-posed in if the following coercivity inequality is satisfied:where is independent of and .

If we set equal to , then we can establish the following coercivity inequality.

Theorem 1.1. Suppose Then the boundary value problem (1.1) is well-posed in a Hölder space and the following coercivity inequality holds: Here is independent of , and

The proof of this assertion follows from the scheme of the proof of the theorem on well-posedness of paper [5] and is based on the following formulas:for the solution of problem (1.1) and on the estimates

Remark 1.2. The nonlocal boundary value problem for the elliptic-parabolic equation in a Hilbert space with a self-adjoint positive definite operator is considered in paper [6]. The well-posedness of this problem in Hölder spaces without a weight was established under the strong condition on .

Now, the applications of this abstract results are presented.

First, the mixed boundary value problem for the elliptic-parabolic equationsis considered. Problem (1.12) has a unique smooth solution for and the smooth functions and This allows us to reduce the mixed problem (1.12) to the nonlocal boundary value problem (1.1) in the Hilbert space with a self-adjoint positive definite operator defined by (1.12).

Theorem 1.3. The solutions of the nonlocal boundary value problem (1.12) satisfy the coercivity inequality where is independent of , and .

The proof of Theorem 1.3 is based on the abstract Theorem 1.1 and the symmetry properties of the space operator are generated by problem (1.12).

Second, let be the unit open cube in the -dimensional Euclidean space with boundary . In the boundary value problem for the multidimensional elliptic-parabolic equationis considered. Problem (1.14) has a unique smooth solution for and , the smooth functions. This allows us to reduce the mixed problem (1.14) to the nonlocal boundary value problem (1.1) in the Hilbert space of all the integrable functions defined on equipped with the normwith a self-adjoint positive definite operator defined by (1.14).

Theorem 1.4. The solution of the nonlocal boundary value problem (1.14) satisfies the coercivity inequality where is independent of , and .

The proof of Theorem 1.4 is based on the abstract Theorem 1.1 and the symmetry properties of the space operator generated by problem (1.14) and the following theorem on the coercivity inequality for the solution of the elliptic differential problem in

Theorem 1.5. For the solution of the elliptic differential problem the following coercivity inequality holds [7]:

2. The First Order of Accuracy Difference Scheme

Let us associate the boundary-value problem (1.1) with the corresponding first order of accuracy difference scheme A study of discretization, over time only, of the nonlocal boundary value problem also permits one to include general difference schemes in applications if the differential operator in space variables, , is replaced by the difference operators that act in the Hilbert spaces and are uniformly self-adjoint positive definite in for .

Let Then the following estimates are satisfied [8]:Furthermore, for a self-adjoint positive definite operator it follows that the operator is defined on the whole space it is a bounded operator, and the following estimates hold:Here From (2.2) and (2.4), it follows that

Theorem 2.1. For any and the solution of problem (2.1) exists and the following formulas hold: where

Proof. By [8, 9],is the solution of the boundary value difference problemis the solution of the inverse Cauchy problemExploiting (2.12), (2.14), and the formulaswe obtain formulas (2.8) and (2.9). For using (2.8), (2.9), and the formulawe obtain the operator equationThe operatorhas an inverse and the following formulais satisfied. This concludes the proof of Theorem 2.1.

Let be the linear space of mesh functions defined on with values in the Hilbert space Next on we denote by and Banach spaces with the normsThe nonlocal boundary value problem (2.1) is said to be stable in if we have the inequalitywhere is independent of not only but also .

Theorem 2.2. The nonlocal boundary value problem (2.1) is stable in norm.

Proof. By [9],for the solution of the inverse Cauchy difference problem (2.15) andfor the solution of the boundary value problem (2.13). The proof of Theorem 2.2 is based on the stability inequalities (2.24), (2.25), and on the estimatesfor the solution of the boundary value problem (2.1). Estimates (2.26) are derived from formula (2.10) and estimates (2.2), (2.4), (2.7). This concludes the proof of Theorem 2.2.

The nonlocal boundary value problem (2.1) is said to be coercively stable (well-posed) in if we have the coercive inequalitywhere is independent of not only but also .

Since the nonlocal boundary value problem (1.1) in the space of continuous functions defined on and with values in is not well-posed for the general positive unbounded operator and space , then the well-posedness of the difference nonlocal boundary value problem (2.1) in norm does not take place uniformly with respect to . This means that the coercive normtends to as The investigation of the difference problem (2.1) permits us to establish the order of growth of this norm to

Theorem 2.3. Assume that and Then for the solution of the difference problem (2.1) we have the almost coercivity inequality where is independent of not only but also .

Proof. By [9], for the solution of the inverse Cauchy difference problem (2.15) andfor the solution of the boundary value problem (2.13). Then the proof of Theorem 2.3 is based on the almost coercivity inequalities (2.30), (2.31), and on the estimatesfor the solution of the boundary value problem (2.1). The proof of these estimates follows the scheme of papers [8, 9] and relies on formula (2.10) and on estimates (2.2), (2.4), and (2.7). This concludes the proof of Theorem 2.3.

Theorem 2.4. Let the assumptions of Theorem 2.3 be satisfied. Then the boundary value problem (2.1) is well-posed in a Hölder space and the following coercivity inequality holds: where is independent of not only but also and .

Proof. By [8, 9],for the solution of the inverse Cauchy difference problem (2.15) andfor the solution of the boundary value problem (2.13). Then the proof of Theorem 2.4 is based on the coercivity inequalities (2.34), (2.35), and on the estimatesfor the solution of the boundary value problem (2.1). Estimates (2.36) are derived from the formulasfor the solution of problem (2.1) and estimates (2.2), (2.4), and (2.7). This concludes the proof of Theorem 2.4.

Now, the applications of this abstract result to the approximate solution of the mixed boundary value problem for the elliptic-parabolic equation (1.14) are considered. The discretization of problem (1.14) is carried out in two steps. In the first step, the grid setsare defined. To the differential operator generated by problem (1.14) we assign the difference operator by the formulaacting in the space of grid functions satisfying the conditions for all With the help of we arrive at the nonlocal boundary-value problemfor an infinite system of ordinary differential equations.

In the second step problem (2) is replaced by the difference scheme (2.1):Based on the number of corollaries of the abstract theorems given above, to formulate the result, one needs to introduce the space of all the grid functions defined on equipped with the norm

Theorem 2.5. Let and be sufficiently small numbers. Then the solutions of the difference scheme (2.41) satisfy the following stability and almost coercivity estimates: where is independent of and .

The proof of Theorem 2.5 is based on the abstract Theorems 2.2, 2.3, on the estimateas well as the symmetry properties of the difference operator defined by formula (2.39) in along with the following theorem on the coercivity inequality for the solution of the elliptic difference problem in

Theorem 2.6. For the solution of the elliptic difference problem, the following coercivity inequality holds [7]:

Theorem 2.7. Let and be sufficiently small numbers. Then the solutions of the difference scheme (2.41) satisfy the following coercivity stability estimates: where is independent of and

The proof of Theorem 2.7 is based on the abstract Theorem 2.4, the symmetry properties of the difference operator defined by formula (2.39), and on Theorem 2.6 on the coercivity inequality for the solution of the elliptic difference equation (2.45) in .

Note that in a similar manner the difference schemes of the first order of accuracy with respect to one variable for approximate solutions of the boundary value problem (1.12) can be constructed. Abstract theorems given above permit us to obtain the stability, the almost stability, and the coercive stability estimates for the solution of these difference schemes.

Acknowledgment

The authors would like to thank Professor P. E. Sobolevskii (Jerusalem, Israel) for his helpful suggestions to the improvement of this paper.