Sobolev conjugate
The Sobolev conjugate of p for , where n is space dimensionality, is
This is an important parameter in the Sobolev inequalities.
Motivation
A question arises whether u from the Sobolev space belongs to
for some q>p. More specifically, when does
control
? It is easy to check that
the following inequality
(*)
can not be true for arbitrary q. Consider , infinitely differentiable function with compact support. Introduce
. We have that
The inequality (*) for results in the following inequality for
If , then by letting
going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for
,
which is the Sobolev conjugate.
See also
References
- Lawrence C. Evans. Partial differential equations. Graduate Studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN 0-8218-0772-2
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