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Sobolev conjugate : ウィキペディア英語版
Sobolev conjugate
The Sobolev conjugate of ''p'' for 1\leq p , where ''n'' is space dimensionality, is
: p^
*=\frac>p
This is an important parameter in the Sobolev inequalities.
==Motivation==
A question arises whether ''u'' from the Sobolev space W^(\mathbb^n) belongs to L^q(\mathbb^n) for some ''q''>''p''. More specifically, when does \|Du\|_ control \|u\|_? It is easy to check that
the following inequality
:\|u\|_\leq C(p,q)\|Du\|_ (
*)
can not be true for arbitrary ''q''. Consider u(x)\in C^\infty_c(\mathbb^n), infinitely differentiable function with compact support. Introduce u_\lambda(x):=u(\lambda x). We have that
:\|u_\lambda\|_^q=\int_|u(\lambda x)|^qdx=\frac\int_|u(y)|^qdy=\lambda^\|u\|_^q
:\|Du_\lambda\|_^p=\int_|\lambda Du(\lambda x)|^pdx=\frac\int_|Du(y)|^pdy=\lambda^\|Du\|_^p
The inequality (
*) for u_\lambda results in the following inequality for u
:\|u\|_\leq \lambda^C(p,q)\|Du\|_
If 1-n/p+n/q\not = 0, then by letting \lambda going to zero or infinity we obtain a contradiction. Thus the inequality (
*) could only be true for
:q=\frac,
which is the Sobolev conjugate.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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