1

Let $$X=\{u\in W^{1,2}(-1,1)\mid u(-1)=u(1), \int_{-1}^1 u=0\}.$$ I proved Wirtinger inequality i.e. $$\int_{-1}^1 (u')^2\geq \pi^2\int_{-1}^1 u^2$$ for all $u\in X\cap \mathcal C^2([-1,1])$. But I want to prove it when $u\in X$ (and not $X\cap \mathcal C^2([-1,1])$). In my course, it's written that $X\cap \mathcal C^2([-1,1])$ is dense in $X$, i.e. that for all $u\in X$ there is a sequence $u_n\in X\cap \mathcal C^2([-1,1])$ s.t. $u_n\to u$ in $W^{1,2}(-1,1)$. Is it a famous result ? Can someone give me a link where such density is mentioned ?

user330587
  • 1,624

1 Answers1

1

Typically, density of $C^2([-1,1])$ in $W^{1,2}(-1,1)$ is shown by considering a mollification $u *\phi_n$ of $u\in W^{1,2}(-1,1)$. So, one way to show what you are asking is to revisit this proof, and actually show that $u * \phi_n\in X$ whenever $u\in X$.

detnvvp
  • 8,237
  • Thanks for your answer. Is there a general result like $\mathcal C^k(a,b)$ dense in $W^{1,p}(a,b)$ or $\mathcal C^k(\Omega )$ dense in $W^{1,p}(\Omega )$ for a bounded open $\Omega \subset \mathbb R^n$ ? – user330587 Jun 02 '17 at 17:43
  • @user330587: see (https://en.wikipedia.org/wiki/Sobolev_space#Approximation_by_smooth_functions) or google "Meyers-Serrin Theorem". – Giovanni Jun 02 '17 at 18:30