Solving differential equation by weak formulation and minimizing a functional

Question

I want to give a weak formulation of the boundary value problem \begin{align*} -(c(x)(u'(x)-1))' & = 0 \textrm{ on } \Omega = (-1,1) \\ u(-1) = u(1) & = 0 \end{align*} where $c(x)$ is piecewise linear. Then solve the weak problem for $$ c(x) = \left\{ \begin{array}{ll} \cosh(\pi) & -1 < x < 0 \\ \sinh(\pi) & 0 \le x < 1 \end{array}\right. $$ I have some questions on my approach, and at a few points I am totally unsure if I am on the right track. First for the weak formulation I define the functional $$ a(u,v) := \int_{-1}^1 -(c(x)(u'(x)-1))' v(x) d x $$ on the space of all continuous functions with $v(-1) = v(1) = 0$. Then by partial integration I get $$ a(u,v) = \int_{-1}^1 c(x)(u'(x)-1) v'(x) d x. $$ But this is not symmetric? In my notes the functional has to be symmetric? Okay, but to proceed, the weak formulations looks for a function $u$ such that $$ a(u,v) = 0 $$ for all $v$ with $v(-1) = v(1) = 0$. Okay I consider the functional $$ E(v) = \frac{a(v,v)}{2} = \int_{-1}^1 c(x)(v'(x) - 1)v'(x) d x. $$ This functional has to be minimized now. To solve this, I should also suppose that $u$ is continuous and linear on $[-1,0]$ and linear on $[0,1]$, this means that $u$ has the form \begin{align*} u(x) & = a x + b \mbox{ for } x \in [-1,0] \\ u(x) & = c x + d \mbox{ for } x \in [0,1]. \end{align*} So this means I have to minimize $E(u)$ over the space of all such functions with $u(-1) = u(1)$? Is this all right? And what about the fact that $a(u,v)$ is not symmetric?

Answer 1

$$ -(c(x)(u'(x)-1))' =0$$ $$ c(x)u''(x) =0 $$

Multiply both sides by $v(x)\in H^1_0([-1,1])$ and integrate by parts: $$ \int_{-1}^{1} c(x)u''(x) v(x)dx=0$$ $$ [c(x)v(x)u'(x)]_{-1}^{1}-\int_{-1}^{1} c(x)u'(x) v'(x)dx=0$$ Due to essential boundary conditions the boundary term vanishes; leaving behind. $$ \int_{-1}^{1} c(x)u'(x) v'(x)dx=0$$ which of the weak form $A(u,v)=F(v)$. It is possible to argue that this is equivalent to the form where we seek $u(x)$ which minimizes $\frac{1}{2}A(u,u)-F(u)$ using the properties of coercivity of A, continuity of A and F, bilinearity of A, linearity of F, and symmetry of A.