I am reading the paper "An Introduction to the Adjoint Approach to Design"(https://link.springer.com/article/10.1023/A:1011430410075). In this paper, derivation of the adjoint equation for the one-dimensional convection-diffusion equation is illustrated. But I don't understand why two terms are zero in the derivation.
The one-dimensional convection-diffusion equation $$ Lu\equiv \frac{du}{dx}-\epsilon\frac{d^2u}{dx^2}, \;\; 0<x<1 $$ subject to the homogeneous boundary conditions $u(0)=u(1)=0$
Using integration by parts, for any twice-differentiable function $v$ we have $$ (v,Lu)=\int_{0}^{1} v \Big(\frac{du}{dx}-\epsilon\frac{d^2u}{dx^2}\Big)dx \\ =\int_{0}^{1} u \Big(-\frac{dv}{dx}-\epsilon\frac{d^2v}{dx^2}\Big)dx + \Big[vu -\epsilon v\frac{du}{dx}-\epsilon u\frac{dv}{dx} \Big]^1_0 \\ =\int_{0}^{1} u \Big(-\frac{dv}{dx}-\epsilon\frac{d^2v}{dx^2}\Big)dx + \Big[\epsilon v\frac{du}{dx} \Big]^1_0 $$
Why $vu$ and $-\epsilon u\frac{dv}{dx}$ are 0 and you get the below? $$ \Big[vu -\epsilon v\frac{du}{dx}-\epsilon u\frac{dv}{dx} \Big]^1_0 = \Big[\epsilon v\frac{du}{dx} \Big]^1_0 $$
The terms $vu$ and $-\epsilon u \frac{dv}{dx}$ are 0 because they are evaluated at the boundaries where $u(0) = u(1) = 0$. The remaining boundary terms prescribes the boundary condition on the costate variable, $v(0) = v(1) = 0$.
I hope that helps.