Exact constants in Poincare type inequalities for functions with zero   mean boundary traces
  In the paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We derive exact and easily computable constants for some basic domains (rectangles, cubes, and right triangles). In the last section, we derive an a estimate of the difference between the exact solutions of two boundary value problems. Constants in Poincare type inequalities enter these estimates, which provide guaranteed a posteriori error control. 
Alexander I. Nazarov, Sergey I. Repin
math.AP