0

Let $\Omega \subset \mathbb R^n$ open with Lipschitz boundary. Let $u\in W^{2,2}(\Omega )$. In a course I have that : by the fundamental lemma of variation calculus, if $$\int_\Omega (\Delta u)\varphi=0$$ for all $\varphi\in W_0^{1,2}$ then $\Delta u=0$ a.e.

I would agree if $\varphi\in \mathcal C_c^\infty (\Omega )$, but here $\varphi\in W_0^{1,2}(\Omega )$ not $\mathcal C_c^\infty (\Omega )$. So why is this true ?

user349449
  • 1,577
  • If it holds for all $H^1_0$ then it holds for all $C_c^\infty$. – math_guy Jun 08 '17 at 18:44
  • @math_guy: I agree but my question is why the fact that it hold in $\mathcal C_c^\infty $ then it hold in $H_0^1$ ? – user349449 Jun 08 '17 at 18:50
  • 2
    you're saying it holds for all $H^1_0$ functions. Since $C_c^\infty$ is a subset of $H^1_0$ it also holds for all $C_c^\infty$ functions, and there you agree that $\Delta u = 0$. Did I miss something? – math_guy Jun 08 '17 at 20:12

0 Answers0