On an estimate for semi-linear elliptic differential equations of the second order

Concerning the estimate of this sort, Nagumo obtained a result on the assumption that the coefficients aτj(x) satisfy the Lipschitz condition ([4], 2) pp. 211-215, Theorem 2), and thereafter Simoda [5] and the author [3] improved Nagumo's result on the assumption that the coefficients ctij(x) satisfy the Holder condition. As, however, it is desirable from a theoretical point of view, that we have the a priori estimate under the weakest possible condition on the continuity of the coefficients #*/#)» we shall form, in this paper, an a priori estimate of the same type as obtained in the above-cited papers, provided that the coefficients aij(x) satisfy the Dini condition. The Dini condition which we impose on the coefficients aτj(x\ is more restrictive than usual, but it seems to be considerably general. Our method of proof in this paper is analoguous to one in the previous paper [3], which was composed of Nagumo's one and Cordes' modified results [2]. Therefore, the parts of the proof which can be carried out in the same way as in the previous paper, will often be omitted. In §2, we shall give two definitions concerning Dini functions, and prove two lemmas in regard to the properties of Dini functions given in these two definitions. In § 3, we state the main result of this paper, whose proof is left to § 5. A set of lemmas will be made in §4, and two other results will be proved in §6.