## Positivity for a strongly coupled elliptic system by Green function estimates.(English)Zbl 0792.35048

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.
Reviewer: G.Sweers (Delft)

### MSC:

 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35B50 Maximum principles in context of PDEs 45M20 Positive solutions of integral equations 35B45 A priori estimates in context of PDEs
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### References:

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