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.
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.
This result is a special case of what is known as Friedrich's Inequality, also sometimes known as Poincare's inequality. It is always true with $n$ arbitrary and $p = q$, although $C$ will depend on $n$ and the size and shape of the support of $u$. However, if you fix the supports of all your $u$'s to lie inside of some fixed set $\Omega$ (i.e. $ u \in C_c(\Omega)$), then you can choose $C$ depending only on $n$ and $\Omega$. Of course, since the support is compact and hence of finite Lebesgue measure, the inequality will also be true with $q \geq p$ simply because in this case $\|f\|_p \le C\|f\|_q$ by Holder's inequality. Any reasonable book on PDEs/Sobolev Spaces should have a proof of Friedrich's inequality, although its not too hard to cook one up yourself. Hint: integration by parts.