A priori estimates for continuation problems for elliptic and principally normal differential equations

Abstract

Introduction. There are well-known procedures due to Douglas [6], [7] for finding an approximation to an analytic or harmonic function in the unit disk when data are given at a finite number of points inside. The success of the error analysis depends on estimates which show that an analytic or harmonic function defined and bounded on the unit disk depends continuously on its restriction to a concentric disk of smaller radius. In general, the problem of determining a function in a given class from its restriction is called the continuation problem. When the function exists, is unique and depends continuously on its restriction, the continuation problem is said to be wellposed in the sense of Hadamard. For practical purposes continuous dependence must be taken to mean Holder continuous dependence. When there is Holder continuous dependence, the continuation problem, in the terminology of F. John, is said to be well behaved. The estimates used in the error analysis of Douglas's numerical procedures, due to Hadamard [12] for analytic functions and to Miller [16] for harmonic functions, are an example of this kind. They show that the continuation problem is well behaved for bounded analytic functions and for bounded harmonic functions in the plane. A more general result is due to F. John, who has shown in [14] that the continuation problem for solutions of linear analytic elliptic equations of arbitrary order in any number of variables is well behaved if a bound on the solutions is prescribed. The subject of this paper is the nature of the continuation problem for the solutions of nonanalytic elliptic equations in more than two variables. The main result is that the continuation problem is well behaved for bounded solutions of (a class of) elliptic equations with C1 coefficients and of arbitrary order. Continuous dependence of the solution on its restriction is expressed in terms of the uniform norm.