Ladyzhenskaya's inequality

Let ${\displaystyle \mathrm{\Omega }}$ be a Lipschitz domain in ${\displaystyle {\mathbb{R}}^n}$ for ${\displaystyle n=2\text{ or }3}$ and let ${\displaystyle u:\mathrm{\Omega }\to \mathbb{R}}$ be a weakly differentiable function that vanishes on the boundary of ${\displaystyle \mathrm{\Omega }}$ in the sense of trace (that is, ${\displaystyle u}$ is a limit in the Sobolev space ${\displaystyle H^1(\mathrm{\Omega })}$ of a sequence of smooth functions that are compactly supported in ${\displaystyle \mathrm{\Omega }}$). Then there exists a constant ${\displaystyle C}$ depending only on ${\displaystyle \mathrm{\Omega }}$ such that, in the case ${\displaystyle n=2}$:

${\displaystyle \Vert u{\Vert }_{L^4}\le C\Vert u{\Vert }_{L^2}^{1/2}\Vert \mathrm{\nabla }u{\Vert }_{L^2}^{1/2}}$

and in the case ${\displaystyle n=3}$:

${\displaystyle \Vert u{\Vert }_{L^4}\le C\Vert u{\Vert }_{L^2}^{1/4}\Vert \mathrm{\nabla }u{\Vert }_{L^2}^{3/4}}$

Generalizations

• Both the two- and three-dimensional versions of Ladyzhenskaya's inequality are special cases of the Gagliardo–Nirenberg interpolation inequality

${\displaystyle \Vert u{\Vert }_{L^p}\le C\Vert u{\Vert }_{L^q}^{\alpha }\Vert u{\Vert }_{H_0^s}^{1-\alpha },}$

which holds whenever

${\displaystyle p>q\ge 1,s>n(\frac{1}{2}-\frac{1}{p}),\text{ and }\frac{1}{p}=\frac{\alpha }{q}+(1-\alpha )(\frac{1}{2}-\frac{s}{n}).}$

Ladyzhenskaya's inequalities are the special cases ${\displaystyle p=4,q=2,s=1}$ ${\displaystyle \alpha =\frac{1}{2}}$ when ${\displaystyle n=2}$ and ${\displaystyle \alpha =\frac{1}{4}}$ when ${\displaystyle n=3}$.

• A simple modification of the argument used by Ladyzhenskaya in her 1958 paper (see e.g. Constantin & Seregin 2010) yields the following inequality for ${\displaystyle u:{\mathbb{R}}^2\to \mathbb{R}}$, valid for all ${\displaystyle r\ge 2}$:

${\displaystyle \Vert u{\Vert }_{L^{2r}}\le Cr\Vert u{\Vert }_{L^r}^{1/2}\Vert \mathrm{\nabla }u{\Vert }_{L^2}^{1/2}.}$

• The usual Ladyzhenskaya inequality on ${\displaystyle {\mathbb{R}}^n,n=2\text{ or }3}$, can be generalized (see McCormick & al. 2013) to use the weak ${\displaystyle L^2}$ "norm" of ${\displaystyle u}$ in place of the usual ${\displaystyle L^2}$ norm:

${\displaystyle \Vert u{\Vert }_{L^4}\le \{\begin{array}{ll}C\Vert u{\Vert }_{L^{2,\mathrm{\infty }}}^{1/2}\Vert \mathrm{\nabla }u{\Vert }_{L^2}^{1/2},& n=2,\\ C\Vert u{\Vert }_{L^{2,\mathrm{\infty }}}^{1/4}\Vert \mathrm{\nabla }u{\Vert }_{L^2}^{3/4},& n=3.\end{array}}$