2
$\begingroup$

If I have an inequality: $\lVert u\rVert_{L^p(R^n)} \le C\lVert\nabla u\rVert_{L^q(R^n)}$ , where $C \in (0,\infty)$ and $u \in C_c^1(R)$, is there a relation between $p, q, n$ such that the inequality holds ?

Thank you for your help.

  • 0
    Usually such questions can be answered quickly by means of a dimensional analysis.2012-05-30
  • 0
    @GiuseppeNegro: is it true that the follwoing condition should hold $q=(np)/(n-p)$ ? Is it sufficient ?2012-05-30
  • 0
    [Gagliardo–Nirenberg–Sobolev inequality](http://en.wikipedia.org/wiki/Sobolev_inequality#Gagliardo.E2.80.93Nirenberg.E2.80.93Sobolev_inequality)2012-05-30
  • 0
    Here is the 'dimensional analysis' I was referring to: [Remark 10](http://books.google.it/books?id=GAA2XqOIIGoC&lpg=PP1&hl=it&pg=PA278#v=onepage&q&f=false). The text refers to it as a 'scaling argument'.2012-06-03

1 Answers 1