Neumann boundary condition

A Neumann boundary condition prescribes the outward normal derivative of the unknown function on the boundary.

Initial data whose normal derivative is nonzero at a wall cannot satisfy homogeneous Neumann boundary conditions continuously all the way to the initial time. For example, $-\mathrm{erf}(ay)$ has wall derivative $-2a{e}^{-a^2}/\sqrt{\pi }$ at $y=\pm 1$, which is nonzero for finite $a>0$. A parabolic mild solution of an abstract Cauchy problem can still take those initial data in an integral or square-integrable sense and satisfy the boundary condition for every positive time. An exactly compatible smooth profile is needed for a classical solution on the closed time interval.

A Neumann eigenfunction is an eigenfunction of a spatial differential operator whose normal derivative vanishes on the boundary. A Sturm-Liouville eigenfunction expansion in these functions gives a heat kernel for homogeneous Neumann boundary conditions.