Coupled boundary condition

A linear boundary condition can relate traces at different endpoints rather than prescribe data independently at each endpoint. For a second-order differential equation, examples are $a_{1}y(a)+b_1{y}^{\prime }(a)+c_{1}y(b)+d_1{y}^{\prime }(b)=0$ together with another independent relation. The resulting operator domain determines whether the boundary form vanishes. Coupling by itself guarantees neither self-adjointness nor its failure.