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This paper deals with the singular perturbation boundary value problem r e(k(v(s))v'{s))' + (sg(v(s)) <p(v{s)))v'(s) + = 0 in R, 1 u(—oc) = A , v(+oo ) = B; e>0, A<B whose solution ve(s) is constructed by the aid of the solution we(t) to the two-point boundary value problem /w\t) (p{t) + ef(t)k(t)/w{t) V _ ek(t) \ g(t) J w(t) ' w{A) = 0, w(B) 0. The restrictions on <p(t), g(t), k(t), and f(t) not only ensure that the two-point boundary value problem has a solution we(t) but also guarantee that as e tends to zero the solution w£(s) pointwise converges to v0(s) = A + (B A)H (s , seR, the solution to the reduced problem, where H(s) is the multiple-valued Heaviside function, G(t) =: f'Ag{s)ds, and &(t) =: fA(p{s)ds. Moreover, the function ue(x, t) =: vg(x/t), as a solution to the Riemann problem 8C(«) + =/M + 9 (kM^\ xeR_ (>0i dt dx t dx V dx J u(x, 0) = A + (B A)H(x), for xeR pointwise converges to u0(x, /) =: v0(x/t), the discontinuous solution of the Riemann problem for the scalar conservation law (e = 0). Obviously, u0{x, t) satisfies the classical Rankine-Hugoniot condition on the line of discontinuity x = tQ>(B)/G(B), and the restriction on <J>(w) and G(u), O(k) G(u)<f>(B)/G{B) > 0 on [A , B], Received February 5, 1991. 1991 Mathematics Subject Classification. Primary 34B, 34D15, 35L65. ©1992 Brown University 547 548 JIANG JIE and WANG XUEKONG is exactly the E-Condition proposed first by Oleinik. The technical arguments, which involve only the use of the Schauder Fixed Point Theorem and integral representations, are elementary.

A singular perturbation nonlinear boundary value problem and the 𝐸-condition for a scalar conservation law