Let $u\in H^1_0(a,b)$(Sobolev space), $p\in C^1[a,b], r\in C[a,b]$, with the latter two being strictly positive.
It is known that: $\Vert u \Vert_{\mathscr A } = \left[\mathscr{A}(u,u)\right]^{\frac{1}{2}}=\int_a^b\left[ p(x)\left(\frac{\partial u}{\partial x}\right)^2 + r(x)\left(u(x) \right)^2\,dx \right]^{\frac{1}{2}}$. This is the definition of the energy norm.
I would like to prove that: If $\Vert u \Vert_{\mathscr A } = 0\Rightarrow u = 0 $ in $H^1_0(a,b)$
If $\|u\|_\mathscr{A}=0$, we must have that the integrand $(p(u')^2+ru^2)^{1/2}=0$ almost everywhere. This further implies that $p(u')^2+ru^2=0$ a.e. Since $p,r>0,$ we must have that each individual term in that sum is $0$ a.e., and in particular $ru^2=0$ a.e. Once again referring to the sign of $r$, we conclude finally that $u=0$ a.e.