Nonlinear Analysis: Theory, Methods & Applications
Nonlinear elliptic systems with variable boundary data
Introduction
Let be a bounded domain with a Lipschitzian boundary Γ, where n⩾2. In this paper, we consider a boundary value problem of the formwhere z and ϕ1 belongs to H1 and H1/2, respectively (for details, see Section 2).
In the case n=3, the above boundary value problem possesses a natural physical interpretation, namely: suppose that is a three-dimensional body with some heaters distributed on the boundary . Let ϕ1=ϕ1(x) denote the temperature at the point . Then the distribution of temperature inside is given by a solution z1=z1(x), of the boundary value problem , (cf. [3]).
Suppose that we change the boundary value data ϕ1 on ϕ2. The first question which arises in this situation is the problem of the existence of a solution of Eq. (1.1) with the new boundary condition z(x)=ϕ2(x) for . If the solution z2=z2(x), z2=ϕ2 on , does exist, we can consider the question of stability of the system. This paper is devoted, first of all, to the stability problem of the systems of form , i.e. we prove that z2(·) tends to z1(·) provided ϕ2 tends to ϕ1 in appropriate topologies.
Following Hadamard, Hilbert and Courant, we say that a given mathematical model of physical, chemical, technical (and the like) phenomena is well-posed if this problem possesses a solution which continuously changes together with the variable boundary data (cf. [3], [5], [10]).
Courant and Hilbert in their monograph write: “A mathematical problem which is to correspond to physical reality should satisfy the following basic requirements: (1) The solution must exist. (2) The solution should be uniquely determined. (3) The solution should depend continuously on the data (requirement of stability)” and, next, they write: “The third requirement, particularly incisive, is necessary if the mathematical formulation is to describe observable natural phenomena. Data in nature cannot possibly be conceived as rigidly fixed: the mere process of measuring them involves small errors …” (cf. [3, Vol II, Chapter III, Section 6.2]).
A problem is said to be ill-posed if does not possess at least one of properties (1)–(3). However, the theory of ill-posed problems pays most attention to the third requirement. Hadamard gave a simple example of an ill-posed initial value problem for partial differential equations. Namely, consider the Laplace equation zxx+zyy=0, x∈(0,π), y∈(−1,1) with the initial conditions , , z(·,·)∈C2. By a direct inspection and Carleman's theorem, we can show that the function is the unique solution of the above problem for Passing with k to infinity, we see that ϕk and ψk tend to null uniformly, but the sequence zk does not converge to the function z0(x,y)=0 which is the unique solution of the Laplace equation with homogeneous initial data. Thus the above initial value problem is ill-posed. Similarly, it is easy to notice that the two-point boundary value problem , x(0)=0, x(π)=ε possesses a solution , , for ε=0, but if we change the boundary condition x(π)=ε, ε≠0 a little, then a solution does not exist. Thus, this problem is ill-posed.
The question of the existence of a solution for the second-order differential equations with boundary data of the Dirichlet, periodic, von Neumann, etc. type was considered in many papers and monographs based on topological and variational methods (see [4], [11], [12], [15], [19] and the references therein). The literature on stability issues for elliptic systems is not very extensive. The problem of the continuous dependence of solutions of a scalar second-order elliptic equations with two-point boundary conditions was considered in the seventies in papers (see, for instance, [6], [7], [16] and the references in these papers). The question of stability of linear partial elliptic equation with variable boundary data was investigated in [8], [14]. All these works deal with scalar systems and are based on some direct methods. The multi-dimensional Dirichlet problem for an ordinary differential system with variable boundary data was investigated by means of a variational method in [18].
In this paper, we consider an elliptic systemwith the boundary data(System , is a particular case of , with z=(−u,v) and ϕ1=(gk,hk).) Basing ourselves on some variational methods and the well-known Ky-Fan theorem, we prove that boundary value problem , possesses at least one solution which continuously depends on boundary data. For given boundary data (gk,hk), denote by (uk,vk) a solution of system (1.3). Without going into details, the main result of this paper is as follows: if the sequence (gk,hk), k=1,2,…, tends to (g0,h0) in the norm topology of H1/2×H1/2, then the sequence (uk,vk) tends to (u0,v0) in the strong topology of H1×H1. In other words, we have proved that boundary value problem , is well-posed.
As far as we know, the question of the continuous dependence on variable boundary data of the solutions of nonlinear partial differential elliptic systems defined on the spaces H1 and H1/2 has not been considered up to now.
This work is divided into three sections. In the first one, we recall some definitions and give the formulation of the main problem. In the second, some auxiliary lemmas are proved. The main results of this paper and examples are presented in the third section.
Section snippets
Formulation of the problem
By , , r>0 (shortly, , we shall denote the Sobolev space of functions u=u(x) defined on a bounded domain , n⩾2, such that , whose (distributional) derivatives ∇u are elements of the space with the normBy we denote the space of all functions for whichequipped with the norm(cf. [1, Theorem 7.48] or [9, Definition 6.8.2]).
Covering
Auxiliary lemmas
Lemma 3.1 On the existence of saddle points If conditions (1)–(4) are satisfied and the functional Fg,h(·,·) is defined by (2.3), then: for any (g,h)∈W, there exists at least one saddle point (y,z)=(yg,h(·),zg,h(·))∈H0, there exist some balls and , such that (yg,h,zg,h)∈By(r1)×Bz(r2) for any (g,h)∈W.
If, additionally, we assume that the functional Fg,h(·,·) is strictly concave-strictly convex, then the saddle point is uniquely determined. Proof By assumption (2a), for any
Continuous dependence on boundary data
In this part, we shall prove some sufficient conditions for the continuous (or semicontinuous) dependence on boundary data of the solutions of variational (2.2) and boundary (2.1) value problems. We shall prove the following: Theorem 4.1 If conditions (1)–(4) are satisfied, (gk,hk)→(g0,h0) in the strong topology of , then is a nonempty set and in the weak topology of H0.
In other words, the set-valued mapping (gk,hk)→Vk is u.s.c. with respect to the strong topology of
References (19)
Dependence of solutions on boundary conditions for second order ordinary differential equations
J. Differential Equations
(1970)Sobolev Spaces
(1975)- et al.
Set-Valued Analysis
(1990) - R. Courant, D. Hilbert, Methods of Mathematical Physics, Wiley-Interscience, New York,...
Partial Differential Equations
(1998)- J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, New York,...
Continuous dependence on parameters and boundary data for nonlinear two-point boundary value problems
Pacific J. Math.
(1972)- et al.
Continuous dependence of the solutions of elliptic boundary value problems on the coefficients, right hand sides and boundary conditions
Quaestiones Math.
(1980/81) - et al.
Function Spaces
(1977)
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