Sobolev conjugate explained

The Sobolev conjugate of p for

1 \leq p <n , where n is space dimensionality, isp ^{* = (pn)/(n - p)} >p This is an important parameter in the Sobolev inequalities.

Motivation

W ^{1,p} (\R ^{n)} belongs to L ^{q(\R} ^{n)} for some q > p. More specifically, when does \|Du\| _{L ^{p(\R} ^{n)}} control \|u\| _{L ^{q(\R} ^{n)}} ? It is easy to check that the following inequality\|u\| _{L ^{q(\R} ^{n)}} \leq C(p,q)\|Du\| _{L ^{p(\R} ^{n)}} (*)

can not be true for arbitrary q. Consider

u(x)\in C ^{n)} _{c(\R} , infinitely differentiable function with compact support. Introduce u _{λ(x):=u(λ} x) . We have that:\begin{align} \|u _{λ} \| ^{q} _{L ^{q(\R} ^{n)}} & = \int _{\R ^{n}} |u( λ x)| ^{qdx = (1)/(λ ^{n})} \int _{\R ^{n}} |u(y)| ^{qdy=λ} ^{-n} \|u\| ^{q} _{L ^{q(\R} ^{n)}} \\ \|Du _{λ\|} ^{p} ^{L ^{p(\R} ^{n)}} & = \int _{\R ^{n}} | λ Du( λ x)| ^{pdx = (λ ^{p})/(λ ^{n})} \int _{\R ^{n}} |Du(y)| ^{pdy=λ} ^{p-n} \|Du \| ^{p \end{align}} _{L ^{p(\R} ^{n)}}

The inequality (*) for

u _{λ} results in the following inequality for u \|u\| _{L ^{q(\R} ^{n)}} \leq λ ^{1- (n)/(+ (n)/(q) p)} C(p,q)\|Du\| _{L ^{p(\R} ^{n)}}

If

1- (n)/(+ p) (n)/(q) ≠ 0 , then by letting λ going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true forq = (pn)/(n - p) ,

which is the Sobolev conjugate.

See also

• Sergei Lvovich Sobolev

References

• Lawrence C. Evans. Partial differential equations. Graduate Studies in Mathematics, Vol 19. American Mathematical Society. 1998.