Sobolev conjugate

The Sobolev conjugate of p for [math]\displaystyle{ 1\leq p \lt n }[/math], where n is space dimensionality, is

[math]\displaystyle{ p^*=\frac{pn}{n-p}\gt p }[/math]

This is an important parameter in the Sobolev inequalities.

Motivation

A question arises whether u from the Sobolev space [math]\displaystyle{ W^{1,p}(\R^n) }[/math] belongs to [math]\displaystyle{ L^q(\R^n) }[/math] for some q > p. More specifically, when does [math]\displaystyle{ \|Du\|_{L^p(\R^n)} }[/math] control [math]\displaystyle{ \|u\|_{L^q(\R^n)} }[/math]? It is easy to check that the following inequality

[math]\displaystyle{ \|u\|_{L^q(\R^n)}\leq C(p,q)\|Du\|_{L^p(\R^n)} \qquad \qquad (*) }[/math]

can not be true for arbitrary q. Consider [math]\displaystyle{ u(x)\in C^\infty_c(\R^n) }[/math], infinitely differentiable function with compact support. Introduce [math]\displaystyle{ u_\lambda(x):=u(\lambda x) }[/math]. We have that:

[math]\displaystyle{ \begin{align} \|u_\lambda \|_{L^q(\R^n)}^q &= \int_{\R^n}|u(\lambda x)|^qdx=\frac{1}{\lambda^n}\int_{\R^n}|u(y)|^qdy=\lambda^{-n}\|u\|_{L^q(\R^n)}^q \\ \|Du_\lambda\|_{L^p(\R^n)}^p &= \int_{\R^n}|\lambda Du(\lambda x)|^pdx=\frac{\lambda^p}{\lambda^n}\int_{\R^n}|Du(y)|^pdy=\lambda^{p-n} \|Du \|_{L^p(\R^n)}^p \end{align} }[/math]

The inequality (*) for [math]\displaystyle{ u_\lambda }[/math] results in the following inequality for [math]\displaystyle{ u }[/math]

[math]\displaystyle{ \|u\|_{L^q(\R^n)}\leq \lambda^{1-\frac{n}{p}+\frac{n}{q}}C(p,q)\|Du\|_{L^p(\R^n)} }[/math]

If [math]\displaystyle{ 1-\frac{n}{p}+\frac{n}{q} \neq 0, }[/math] then by letting [math]\displaystyle{ \lambda }[/math] going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for

[math]\displaystyle{ q=\frac{pn}{n-p} }[/math],

which is the Sobolev conjugate.

See also

• Sergei Lvovich Sobolev
• conjugate index

References

• Lawrence C. Evans. Partial differential equations. Graduate Studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN:0-8218-0772-2