I want to find elements in Sobolev space $W^{1,2}$[0, 1] corresponding to the functional $f(x) = \int_{0}^{1}x(t)dt$ on the same Sobolev space.
My attempt: $f(x) = (x,y) = \int_{0}^{1}(x(t)y(t) + {x}'(t){y}'(t))dt = \int_{0}^{1}x(t)y(t)dt + \int_{0}^{1}{x}'(t){y}'(t)dt = x(1){y}'(1) - x(0){y}'(0) + \int_{0}^{1}{y}'(t)dt = x(1){y}'(1) - x(0){y}'(0) - \int_{0}^{1}x(t){y}''(t)dt + \int_{0}^{1}x(t)(y(t)) - {y}''(t))dt$
Next step, we should solve differential eq according to our functional:
$y(t) - {y}''(t) = 0$
${y}''(t) = 1$
Got stuck in there since I think we need additional condition in our differential eq.
Thank in advance for any help.