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For a smooth incompressible direct trajectory with total velocity field $\mathbf{U}$ and active scalar $\Theta$, the negative-constraint Lagrangian function in constrained optimization convention givesThe transpose term is the formal adjoint of $\delta \mathbf{u}\cdot \nabla \mathbf{U}$; the two couplings transpose scalar transport by advection and buoyancy. For terminal cost $\int \Theta (T{)}^2$, the terminal data are ${\mathbf{u}}^{†}(T)=0$, ${\theta }^{†}(T)=2\Theta (T)$. They are integrated backward along the stored direct trajectory. No-slip boundary conditions and Dirichlet boundary conditions for the adjoint velocity and homogeneous Neumann boundary conditions for the adjoint scalar remove the spatial boundary terms.

For $H=H_0^1(U)\times H^1(U)$, the coupled formsatisfiesThe Poincare inequality for $u$ then controls both $\Vert u{\Vert }_2$ and $\Vert w{\Vert }_2$, so the form is coercive on $H$. The Lax-Milgram theorem therefore gives a unique weak solution. The first component has a Dirichlet boundary condition, while the second component's Neumann boundary condition is the natural boundary condition in the weak formulation.

The formal $L^2$ gradient flow of this energy is $u_t=\mathrm{div}({\Phi }^{\prime }(r)\nabla u/r)$, with a Neumann boundary condition for no flux. The radial Hessian eigenvalues of the gradient integrand are ${\Phi }^{\prime \prime }(r)$ and ${\Phi }^{\prime }(r)/r$. Their signs distinguish convex forward diffusion from formal backward diffusion. The heat equation uses $\Phi (r)=r^2/2$, whereas total variation flow uses $\Phi (r)=r$ with a subgradient interpretation at zero.

As the jump-length weight tends to infinity with the gradient weight fixed, the limiting problem minimizes $\Vert u-g{\Vert }_2^2+\alpha \Vert \nabla u{\Vert }_2^2$ on $H^1$. It has a unique minimizer by the Lax-Milgram theorem and strict convexity. Without prescribed boundary values, its weak equation is $u-\alpha \Delta u=g$ with natural Neumann boundary conditions.

For a fixed sufficiently regular image edge set, the Mumford–Shah functional is a strictly convex quadratic energy on the Sobolev space of its complement. A first variation gives $u-\alpha \Delta u=g$ within each region and separate one-sided Neumann boundary conditions on free edges. A jump need not vanish across an edge. The weak solution is unique by the Lax-Milgram theorem; this does not prove uniqueness when the edge set is also optimized.

Free endpoint variations set the open string flux $e{T}^2{X}_m^{\prime }+u{P}_m$ to zero. In a boundary-adapted temporal gauge, freedom to vary time forces $u=0$ at the end. Nonzero lapse then gives $X^{\prime m}=0$, a Neumann boundary condition.

For the linear dispersive Stokes equation on $x>0$, with sufficiently smooth decaying initial data and $\mathrm{Re}p>0$, define $V(x,p)={\int }_0^{\infty }{R}_p(x-y)u_0(y)dy$ and let $G(p)$ be the time Laplace transform of the prescribed derivative $u_x(0,t)$. Spatial decay and the one-dimensional decaying homogeneous space giveThe Neumann boundary condition follows by differentiating at zero. The initial forcing follows from the Stokes resolvent kernel with advection. The Bromwich inversion formula gives an integral representation with no unknown boundary traces. Truncating or extending the prescribed boundary signal beyond a target time does not affect earlier values, by the Laplace transform time-shift rule and causality of this Green function.

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 linear fluctuating interface smooths height gradients by diffusion and receives additive Gaussian white noise. With periodic or Neumann boundary conditions, its nonzero Fourier modes are independent Ornstein-Uhlenbeck processes if the noise is diagonal in that basis. The unconstrained uniform height is a Brownian zero mode of a fluctuating interface.

The symmetry used to classify an open string must preserve its endpoint boundary background as well as its momentum. A Neumann-Dirichlet open-string boundary condition breaks target Lorentz transformations mixing ND and common directions with Neumann boundary conditions. For a D1–D25 pair with 24 ND directions, the common worldvolume has Lorentz group $SO(1,1)$, whose connected massive little group is trivial. The transverse $SO(24)$ remains as an internal rotation group. Thus a massive 24-component oscillator multiplet is not an $SO(25)$ vector of the full 26-dimensional massive little group; a vector of that full group would have 25 components.

With 24 physical half-integer open-string oscillators, the first three string level operator eigenvalues are $0,1/2,1$. Their states are the vacuum, $a_{1/2}^{i†}|0⟩$, and $a_{1/2}^{i†}{a}_{1/2}^{j†}|0⟩$. The creation operators commute, so the third level is the symmetric square of the 24-dimensional transverse vector, of dimension $24\cdot 25/2=300$. Its trace is a scalar and its trace-free part has dimension 299. Including the vacuum shift gives ${\alpha }^{\prime }{M}^2=1/2,1,3/2$ when the two longitudinal directions are common directions with Neumann boundary conditions. The representation labels refer to the surviving ND rotation group, not full unbroken target Lorentz symmetry.

A mixed boundary condition imposes different kinds of boundary conditions on different portions of the boundary. A common elliptic example combines a Dirichlet boundary condition on one portion with a Neumann boundary condition on the remainder.

A transverse open string coordinate has an ND boundary condition when one endpoint obeys a Neumann boundary condition and the other a Dirichlet boundary condition. On a strip of width $\pi$, its nonconstant spatial eigenfunctions are $\cos ((n+1/2)\sigma )$ when the free endpoint is at zero. They solve the endpoint equations because their derivatives vanish at zero and their values vanish at $\pi$. Reversing the endpoints gives sine eigenfunctions with the same frequencies. The fixed endpoint removes the dynamical worldsheet zero mode.

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.

For the heat equation $u_t=\frac{1}{2}{u}_{xx}$ on $x>0$ with the Neumann boundary condition $u_x(t,0)=0$, the method of images givesThus $u(t,x)={\int }_0^{\infty }{K}_t^N(x,y)f(y)dy$. This is also the transition kernel of Reflected Brownian motion.

For $L>0$ and $t>0$, the Sturm-Liouville eigenfunction expansion for $u_t=u_{xx}$ on $(0,L)$ with homogeneous Neumann boundary conditions givesThe constant eigenfunction mode preserves the spatial integral. This is the $\beta =0$ limit of the weighted Neumann heat kernel with constant drift, and its boundary inputs have the signs in the Neumann boundary-forcing formula.

An open string has a worldsheet with boundary. Its endpoint conditions may be Neumann boundary conditions, Dirichlet boundary conditions, or mixtures selected independently in target-space directions.

In conformal gauge, a free open-string coordinate with Neumann boundary conditions has a cosine expansion. Canonical quantization turns its nonzero Fourier coefficients into string oscillators.