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| In [[mathematical analysis]], '''Trudinger's theorem''' or the '''Trudinger inequality''' (also sometimes called the '''Moser–Trudinger inequality''') is a result of [[functional analysis]] on [[Sobolev space]]s. It is named after [[Neil Trudinger]] (and [[Jürgen Moser]]).
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| It provides an inequality between a certain [[Sobolev space]] norm and an [[Orlicz space]] norm of a function. The inequality is a [[limiting case]] of Sobolev imbedding and can be stated as the following theorem:
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| Let <math>\Omega</math> be a bounded domain in <math>\mathbb{R}^n</math> satisfying the [[cone condition]]. Let <math>mp=n</math> and <math>p>1</math>. Set
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| :<math>
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| A(t)=\exp\left( t^{n/(n-m)} \right)-1.
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| </math>
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| Then there exists the imbedding
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| :<math>
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| W^{m,p}(\Omega)\hookrightarrow L_A(\Omega)
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| </math> | |
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| where
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| :<math>
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| L_A(\Omega)=\left\{ u\in M_f(\Omega):\|u\|_{A,\Omega}=\inf\{ k>0:\int_\Omega A\left( \frac{|u(x)|}{k} \right)~dx\leq 1 \}<\infty \right\}.
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| </math>
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| The space
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| :<math>L_A(\Omega)</math>
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| is an example of an [[Orlicz space]].
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| ==References==
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| *{{citation|last=Moser|first=J.|authorlink=Jürgen Moser|title=A Sharp form of an Inequality by N. Trudinger|journal=Indiana Univ. Math.|volume=20|year=1971|pages=1077–1092}}.
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| *{{citation|last=Trudinger|first=N. S.|authorlink=Neil Trudinger|title=On imbeddings into Orlicz spaces and some applications|journal=J. Math. Mech. |volume=17|year=1967|pages=473–483}}.
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| [[Category:Sobolev spaces]]
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| [[Category:Inequalities]]
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| [[Category:Theorems in analysis]]
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Latest revision as of 06:22, 3 September 2014
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