Neumann boundary condition

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

A symmetric uniformly elliptic divergence-form energy with zeroth-order coefficient at least $\mu >0$ controls the full $H^1$ norm, including constants. Its quadratic energy minus a bounded load has a unique minimizer by the direct method, and vanishing first variation gives the natural Neumann weak equation. Unlike a pure Neumann Laplacian, this problem needs no zero-mean compatibility condition. The exact energy difference from the weak solution is half the bilinear energy of the difference.

If a harmonic function on one side of a straight boundary has zero normal derivative there, even reflection extends it harmonically across the boundary. Locally take the boundary to be $x=0$ and assume $u$ and its first derivatives extend continuously to its open segment. Define $\tilde{u}(x,y)=u(|x|,y)$. The zero value of $u_x$ makes the first derivatives match across the join. Applying Green's first identity on the two sides against a smooth compactly supported test function leaves no interface term, so the reflected function is weakly harmonic and hence harmonic by Weyl lemma.

Reflecting across both edges of a right-angle corner gives a harmonic function on a punctured neighbourhood when both normal derivatives vanish. If the original function is bounded at the corner, the removable singularity for a bounded harmonic function makes the extension harmonic there too. Thus a maximum on such an edge or corner can be tested using the interior strong maximum principle for harmonic functions.

Neumann boundary data are prescribed values of the outward normal derivative of a function on the boundary of a boundary value problem. They enter a Neumann boundary condition as ${\partial }_{n}u=g_N$. Directional derivatives defined using a fixed coordinate direction must be converted to the outward-normal convention: for example, on the bottom of an upper domain, ${\partial }_{n}u=-u_y$.

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.