# EDIT: Estimation of solution of variational problem

We consider the problem $$-u''+a(x)u=f, x\in (0,1)=\Omega , u(0)=\alpha,u(1)=\beta$$ with $f \in L^2(\Omega) , a(x) \geq a_0 > 0, a \in L^{\infty}(\Omega)$

Question 1: prove that the variational formulation associate to this problem admits a unique solution in an adequate Hilbert space $V$ (we can use for this proof the function $u_1(x)=\alpha +(\beta - \alpha) x$).

Question 2: prove that $$||u||_V \leq C ||f||_{L^2}$$

For the question 1, I consider $u=w + u_1$ with $w \in V = H^1(\Omega)$ and solution for the equation of the problem, and I prove that $w$ is unique in $H^1_0(\Omega)$ by Lax-Milgram, so the problem admits a unique solution $u$ in $H^1_0.$

My problem is in question 2, I don't know how we prove it. Thank's for the help.

• Your second question is strange. If $u$ is a solution that belongs to $V$, then it is immediate that $\|u\|_V$ is finite. – Tomás Aug 10 '13 at 20:24