Direct-ethanol fuel cell: Difference between revisions

Revision as of 20:53, 28 February 2013

The Sobolev conjugate of p for $1 \leq p < n$ , where n is space dimensionality, is

p^{*} = \frac{p n}{n - p} > p

This is an important parameter in the Sobolev inequalities.

Motivation

A question arises whether u from the Sobolev space $W^{1, p} (R^{n})$ belongs to $L^{q} (R^{n})$ for some q>p. More specifically, when does $‖ D u ‖_{L^{p} (R^{n})}$ control $‖ u ‖_{L^{q} (R^{n})}$ ? It is easy to check that the following inequality

‖ u ‖_{L^{q} (R^{n})} \leq C (p, q) ‖ D u ‖_{L^{p} (R^{n})}

(*)

can not be true for arbitrary q. Consider $u (x) \in C_{c}^{\infty} (R^{n})$ , infinitely differentiable function with compact support. Introduce $u_{λ} (x) : = u (λ x)$ . We have that

‖ u_{λ} ‖_{L^{q} (R^{n})}^{q} = \int_{R^{n}} | u (λ x) |^{q} d x = \frac{1}{λ^{n}} \int_{R^{n}} | u (y) |^{q} d y = λ^{- n} ‖ u ‖_{L^{q} (R^{n})}^{q}

‖ D u_{λ} ‖_{L^{p} (R^{n})}^{p} = \int_{R^{n}} | λ D u (λ x) |^{p} d x = \frac{λ^{p}}{λ^{n}} \int_{R^{n}} | D u (y) |^{p} d y = λ^{p - n} ‖ D u ‖_{L^{p} (R^{n})}^{p}

The inequality (*) for $u_{λ}$ results in the following inequality for $u$

‖ u ‖_{L^{q} (R^{n})} \leq λ^{1 - n / p + n / q} C (p, q) ‖ D u ‖_{L^{p} (R^{n})}

If $1 - n / p + n / q = 0$ , then by letting $λ$ going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for

q = \frac{p n}{n - p}

,

which is the Sobolev conjugate.

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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+The '''Sobolev conjugate''' of ''p'' for <math>1\leq p <n</math>, where ''n'' is space dimensionality, is
+:<math> p^*=\frac{pn}{n-p}>p</math>
+This is an important parameter in the [[Sobolev inequality|Sobolev inequalities]].
+== Motivation ==
+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
+the following inequality
+:<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
+:<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>
+:<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>
+The inequality (*) for <math>u_\lambda</math> results in the following inequality for <math>u</math>
+:<math>\|u\|_{L^q(R^n)}\leq \lambda^{1-n/p+n/q}C(p,q)\|Du\|_{L^p(R^n)}</math>
+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
+:<math>q=\frac{pn}{n-p}</math>,
+which is the Sobolev conjugate.
+==See also==
+*[[Sergei Lvovich Sobolev]]
+==References==
+* Lawrence C. Evans. Partial differential equations. Graduate studies in Mathematics, Vol 19. American Mathematical Society. 1998.  ISBN 0-8218-0772-2
+[[Category:Sobolev spaces]]

Direct-ethanol fuel cell: Difference between revisions

Revision as of 20:53, 28 February 2013

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