Hi can anyone help me with this question?
Let $u\in C^2(B(0,1))$, where $B(0,1)$ is the unit ball, such that $u$ vanishes at the boundary and let $u$ solve the Schrödinger equation $\Delta u-V(x)u=f(x)$ with $f\in L^\infty$ and $V(x)\ge 1$ such that $V(x)\in L^\infty$.
I now have to find a bound for $||u||_{H^1}$ in terms of $||f||_{L^2}$
We have, with $B=B(0,1)$, multiplying by $u$ both sides of the equation and integrating by parts (recall $u$ vanishes at the boundary) $$ \int_B |\nabla u|^2 +u^2 dx \leq \int_B |\nabla u |^2 dx + V(x)u^2 dx = -\int_B fudx \leq \| u\|_{L^2}\| f\|_{L^2}, $$ where the first inequality is because $V\geq1$ and the last one is Cauchy-Schwarz. Therefore we get $$ \| u\|_{H^1}^2 \leq \| f\|_{L^2}\| u\|_{L^2}\leq \| f\|_{L^2}\| u\|_{H^1}. $$ This is what you want.