What are the optimal constants for the Sobolev inequalities in 3D?

[Gerenuk]

I'm searching for an inequality between
\iiint_\infty |\nabla f|^2 \mathrm{d}^3r
and
\iiint_\infty |f|^2 \mathrm{d}^3r

I saw similar inequalities that they called Sobolev inequalities. What would be the correct form and optimal constant for this 3D case?

 

[maze]

For functions that are zero on a lipschitz boundary, this is Poincare's inequality for W₀^1,2. A straightforward proof is found in Evans PDE page 265. If you want to find the exact constant, you could go through the proofs of the relevant estimates on the previous several pages, carefully keeping track of the constants. I haven't done this. The constant will be dimension-dependent.

For functions that are not zero on the boundary, you have to subtract off the average value,
||u-avg(u)||_L2 <= C||Du||_L2

Which is another "Poincare inequality" (Evand p. 275)