Ladyzhenskaya's inequality
= Ladyzhenskaya's inequality
{c}
{wiki}
<Ladyzhenskaya's inequality> controls the <Lp norm> of a function by its <Lp norm> and the <Lp norm> of its <gradient>: in dimension two, $\|u\|_4\leq C\|u\|_2^{1/2}\|\nabla u\|_2^{1/2}$; in dimension three the exponents are $1/4$ and $3/4$. These estimates hold for smooth compactly supported functions and extend using <density of smooth functions in a Sobolev space>.