A priori estimates for solutions to elliptic equations in long domains
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We consider a second-order linear uniformly elliptic partial differential operator in non-divergence form. For the operator, there is a well-known Aleksandrov-Bakel'man-Pucci estimate(ABP estimate, in short). Following the proof of the original ABP estimate, using a rectangular cone than a circular cone, we obtain a smaller constant than the original estimate for the upper bound. Also, we show that our improved result implies the original ABP estimate and is more useful for long domains than the original one.
Abstract
We consider a second-order linear uniformly elliptic partial differential operator in non-divergence form. For the operator, there is a well-known Aleksandrov-Bakel'man-Pucci estimate(ABP estimate, in short). Following the proof of the original ABP estimate, using a rectangular cone than a circular cone, we obtain a smaller constant than the original estimate for the upper bound. Also, we show that our improved result implies the original ABP estimate and is more useful for long domains than the original one.