410 Proc. Japan Acad., 59, Ser. A (1983) [Vol. 59 (A), 119. Boundary Value Problems for Some Degenerate Elliptic Equations of Second Order By Hirozo OKUMURA Department of Applied Mathematics and Physics, Kyoto University (Communicated by K,Ssaku YOSIDA, M. J. A., Nov. 12, 1983) 1. Let/2 be a bounded domain in R with C boundary 30 and a(x)=a(x), b(x) and c(x) be real valued functions belonging to C(O). in this note we shall consider the regularity up to the boundary of the solution for the following boundary value problem" [P] Au= a’(x) + b(x) u ,= xx x +c(x)u= f(x) in O, u[o,=0, under the assumptions on A" N A1 a(x, )= a’(x)O for (x, ) e 9X (R(0}). % A2 A3 A4 c(x)<0 and c*(x)=c(x)-V. 3b (x)+ ,(x)<O on 9. 3x , 3x3x 39 is non-characteristic for A. (, )= b()- (z) for (x, ) e ={(x, ) e 9X (R{O})la(x, )=0}. Several existence, uniqueness and regularity theorems of the problem [P] were proved in Fichera [1], [2], Kohn-Nirenberg [4] and Oleinik [5], Oleinik-Radkevi5 [6]. In fact, it is known that there is a uniquely determined weak solution u e L(O) of [P] with f e LZ(O) if the conditions A1, A2 and A3 hold. Here u e L(O) is called the weak solution of [P] with f e L(O) if the identity (1.1) uAvdx= fdx holds for all v e C() with v]0=0, where v { - a" }V_+c,(x)v A’v-- aJ(x) bJ(x)--2 }- 3x, (x) t,= axax = Concerning the local regularity of this weak solution, we can apply Theorem 5.9 in HSrmander [3] to the operator A if the condi- tions A1 and A4 hold (see also Radkevi5 [7]). That is, if u is the weak solution of [P] with f e H(O)," then we have u e H+(U) for any open set U such that/7c0. This is the reason why we consider the. boundary value problem [P]. 1) Hk(tg) denotes the Sobolev space on 9 for non-negative integer k.