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| The '''Sobolev conjugate''' of ''p'' for <math>1\leq p <n</math>, where ''n'' is space dimensionality, is
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| :<math> p^*=\frac{pn}{n-p}>p</math>
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| This is an important parameter in the [[Sobolev inequality|Sobolev inequalities]].
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| == Motivation ==
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| A question arises whether ''u'' from the [[Sobolev space]] <math>W^{1,p}(R^n)</math> belongs to <math>L^q(R^n)</math> for some ''q''>''p''. More specifically, when does <math>\|Du\|_{L^p(R^n)}</math> control <math>\|u\|_{L^q(R^n)}</math>? It is easy to check that
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| the following inequality
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| :<math>\|u\|_{L^q(R^n)}\leq C(p,q)\|Du\|_{L^p(R^n)}</math> (*) | |
| can not be true for arbitrary ''q''. Consider <math>u(x)\in C^\infty_c(R^n)</math>, infinitely differentiable function with compact support. Introduce <math>u_\lambda(x):=u(\lambda x)</math>. We have that
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| :<math>\|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</math>
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| :<math>\|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</math>
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| The inequality (*) for <math>u_\lambda</math> results in the following inequality for <math>u</math>
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| :<math>\|u\|_{L^q(R^n)}\leq \lambda^{1-n/p+n/q}C(p,q)\|Du\|_{L^p(R^n)}</math>
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| If <math>1-n/p+n/q\not = 0</math>, then by letting <math>\lambda</math> going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for
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| :<math>q=\frac{pn}{n-p}</math>,
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| which is the Sobolev conjugate.
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| ==See also==
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| *[[Sergei Lvovich Sobolev]]
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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
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| [[Category:Sobolev spaces]]
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