The best constant in Poincare-liked inequality in BVBV and BDBD space

This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention.

Let ΩRN be open, bounded and with smooth boundary. Then we could prove that for any uBV(Ω) and ωkerE, where E denotes the distributional symmetric derivative Eω=12(ω+ωT) from RN to RN, there exists a constant C>0 independent of u and ω such that
DuM(Ω)C(DuωM(Ω)+uL1)
The proof is not long and can be found here, Theorem 3.3, equation (4).

It can also be shown that ωkerE iff ω=Ax+b where A=AT, ARN×N and bRN

However, the proof is done by using contradiction, which is short and simple but can not give any information of this constant C.

I am interested in finding the best constant C>0. I am sure this constant should only depend on Ω but I really want to know how it depends on Ω. Can we make the best constant larger or smaller by changing Ω? Also, if Ω:=[0,1]×[0,1] in R2, can we explicitly compute this constant?

Thank you!

Answer

Attribution
Source : Link , Question Author : JumpJump , Answer Author : Community

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