I am trying to understand the variational method and the connection to the Lax-Milgram-Theorm. I don't know how to use the theory to solve this exercise.
Let $\epsilon > 0$ and we have the boundary value problem $$ -u_{xx} = \epsilon u \,\text{for} \, -1<x<1 \\ u(-1)=u(1)=0$$
Show that you can write this boundary value problem as a variational equation as follows: Let $$ b: H \rightarrow \mathbb{R} $$ a linear functional. Find $u \in H$ such that $$ a(u,v) = b(v) $$ for all $v \in H$. In our case $H:= H_{0}^{1}(-1,1)$ is a Hilbertspace.
Second part of this exercise: Show with the Lax-Milgram-Theorem for which $e>0$ this boundary problem has exactly one solution.
Can somone give me hint or explain me how to solve this?
Hint: Take $a(\cdot,\cdot):H_{0}^{1}(-1,1)\times H_{0}^{1}(-1,1)\rightarrow \mathbb{R} $ defined by $a(u,v)=(u_x,v_x)_{L^2(-1,1)}-\varepsilon(u,v)_{L^2(-1,1)}$.