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Sobolev conjugate

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The Sobolev conjugate of p for , where n is space dimensionality, is

This is an important parameter in the Sobolev inequalities.

Motivation

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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

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References

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