On Hartman-like conditions for second order elliptic systems

P. AMSTER

Guanajuato

Congreso; MCA 2013; 2013

In 1960, Hartman proved the existence of solutions for a system of second order ordinary differential equations x''(t) = f (t, x(t), x (t)) with Dirichlet conditions, under a geometric condition on the nonliner term. Roughly speaking, when f does not depend on x the condition says that the field f (t, x) points outwards a ball B centered in the origin for any x ∈ B; in the general case, it is allowed that f points inward but not too much if the velocity x'(t) is small. In this work, we extend this result in several directions. In first place, we find a suitable extension for elliptic systems. In second place, we replace the ball B by a general smooth domain, and establish a condition based on the use of Gauss second fundamental form.