On generalized Peano's theorem concerning the Dirichlet problem for semi-linear elliptic differential equations
Isamu Hirai, Kiyoshi Akô · Proceedings of the Japan Academy Series A Mathematical Sciences · 1960
The purpose of this note is to prove a theorem which concerns the Dirichlet problem for semi-linear elliptic differential equations and is similar to Peano's theorem concerning the initial value problem of ordinary differential equations of the first order."The precise statement of the theorem will be given in 2.The authors of this note wish to express their deepest gratitude to Prof. Masuo Hukuhara who has constantly inspired and stimulated them.1. Preliminaries.In this note we shall consider the semi-linearxx in a bounded domain G under the following assumptions.Assumptions. 1 .G is a bounded Poincar domain in the Eu- clidean m-space; i.e. for each boundary point x of G there exist one half C of a circular cone with vertex x and a closed sphere Kx with center x such that 2 .The symmetric matrix ][a(x)]] is continuous and positive- definite in the closure G of G.3 .The function f(x, u, p) (P=(Pl,'" ", P)) is defined in 'xeG,]u]< , P l< and HSlder-continuous (with some exponent a, 0< a < 1) in every compact subset of ).Further f(x, u, p) is non-decreas- ing with respect to u; i.e.f (x, u, p) f (x, , p) provided x e G, u < , p < Moreover, we assume that for every constant M0 there exist two constants B(M) and F(M) such that If(x, u, p)[ B(M) p -t-F(M) 1) As for generalized Peano's theorem concerning the Dirichlet problem see T. SatS" Sur l'equation aux derivdes partielles Az---f(x, y, z, p, q) I, Compositio Math., 12, 157- 177 (1954); II, ibid., 14, 152-172 (1959).See, in particular, Thdorme 3 of the second note.2) Here x=(xt,..., x) and grad u=(u/xl,..., u/xn).3) We denote by G the closure G+F of the domain G, where F is the boundary of G.