Positivity for a strongly coupled elliptic system by Green function estimates (Q1323557)

Uniform positivity for elliptic systems on bounded domains that are coupled by first order derivatives (for example \(-\Delta u = f - q \cdot \nabla v\), \(-\Delta v = au\)) cannot be obtained by the classical maximum principle. However, using the pointwise estimates \[ | \int_ \Omega G(x,z)a(z)G(z,y)dz| \leq c_ 1 G(x,z)\text{ and }| \int_ \Omega G(x,z) q(z) \cdot \nabla_ z G(z,y)dz| \leq c_ 2 G(x,z), \] where \(G(\cdot,\cdot)\) denotes the Green function, one may show for the above example that if \(a\) and \(q\) are small enough, then \(f > 0\) implies \(u > 0\). Such estimates are given for \(a\) and \(q\) in appropriate Schechter type spaces. More general systems are considered, both in dimensions \(n = 2\) and \(n \geq 3\). The estimates are also used to obtain pointwise bounds for the lifetime of a conditioned brownian motion.