Nash inequality on a compact domain? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T14:32:43Z http://mathoverflow.net/feeds/question/108713 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108713/nash-inequality-on-a-compact-domain Nash inequality on a compact domain? RadonNikodym 2012-10-03T15:28:11Z 2012-10-17T18:22:00Z <p>I have come across a few papers that make use of the <a href="http://en.wikipedia.org/wiki/Sobolev_inequality#Nash_inequality" rel="nofollow">Nash inequality</a> for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the classical Nash inequality on $\mathbb{R}^n$ to that on compact domains so trivial? </p> <p>I'd really appreciate any references you know of.</p> <p>EDIT: This is the statement I am looking for </p> <p>Let $\mathcal{D} = \mathbb{T}^n$ be the unit square in $\mathbb{R}^n$ with periodic boundary conditions. There exist constants $C_1$ and $C_2$ such that such that for $f \in H^1(\mathcal{D})$ then $$||f||_{2}^{1 + \frac{n}{2}} \leq ||f||_1 \left(C_1||f||_{2}^2 + C_2||\nabla f||_2^2\right)^{\frac{n}{4}}$$</p> http://mathoverflow.net/questions/108713/nash-inequality-on-a-compact-domain/108727#108727 Answer by Luis Silvestre for Nash inequality on a compact domain? Luis Silvestre 2012-10-03T17:40:54Z 2012-10-03T17:40:54Z <p>The original Nash inequality in $\mathbb R^d$ is $$\|\nabla f\|_2 \|f\|_1^{2/d} \geq c \|f\|_2^{1+2/d}$$ It is proved in this article: <em>Nash, J. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 1958 931–954.</em></p> <p>The inequality is proved by the beginning of the paper. You can see it by the top of page 936. It is a very simple argument using the Fourier transform, so you can try to work it out in your periodic setting using Fourier series.</p> <p>Note that the function $f \equiv 1$ fails the original Nash inequality in a compact domain. The extra term you have in yours is a correction for the compact case that is not necessary in the full space $\mathbb R^d$</p>