MONOTONE PROBLEMS IN CYLINDERS WITH VANIS HING DIAMETER

Let a : R(N) --> R(N) be a Lipschitz-continuous and strongly monotone function. Let OMEGA = omega x (0, L) be the cylinder of height L and s ection omega. Let H(D)1 (OMEGA) be the space of those functions of H1 (OMEGA) which vanish on the sections x(N) = 0 and x(N) = L of the cyli nder. We consider the following nonlinear monotone boundary value prob lem: u(epsilon) is-an-element-of H(D)1 (OMEGA), integral(OMEGA) a (D(e psilon) u(epsilon)) D(epsilon) upsilon dx = integral(OMEGA) h(partial derivativeupsilon/partial derivativex(N)) dx, for-all upsilon is-an-el ement-of H(D)1 (OMEGA), where D(epsilon) upsilon = t((1/epsilon) (part ial derivative/partial derivativex1),..., (1/epsilon) (partial derivat iveupsilon/partial derivativex(N-1)), (partial derivativeupsilon/parti al derivativex(N)) for any upsilon : OMEGA --> R. We prove in the pres ent Note that u(epsilon) strongly converges in H(D)1 (OMEGA) to the so lution u is-an-element-of H0(1) (0, L) of a monotone problem defined o n the segment (0, L). We also prove that D(epsilon) u(epsilon) and a ( D(epsilon) u(epsilon)) strongly converge in (L2 (OMEGA))N to some limi ts which are computed explicitly. Finally we give an estimate in epsil on of the convergence rate of these quantities for some particular h.