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For a rigid cylinder of radius $a$ and height $h$, the velocity potential obeys the wave equation with Neumann boundary conditions. Separation in cylindrical coordinates gives radial factors $J_m(k_{m,n}r/a)$, angular sine and cosine factors of integer order $m$, and axial factors $\cos (\ell \pi z/h)$. The nonzero radial numbers satisfy $J_m^{\prime }(k_{m,n})=0$, yielding the displayed frequencies.

There is also a constant radial mode, with $m=0$ and radial number zero. A formal zero of $J_m^{\prime }$ at the origin for higher $m$ must not be counted by substituting $k=0$ into $J_m(kr)$, which would give the zero function. Solving the zero-radial-eigenvalue equation directly leaves only the constant mode under the side-wall condition. The fully constant potential has zero velocity and is a gauge mode; constant radial modes with nonzero axial index are physical longitudinal oscillations.

A separated acoustic wave in a straight rigid channel $0<y<H$ has transverse Neumann boundary conditions, giving factors $\cos (n\pi y/H)$ for integers $n\ge 0$. The velocity potential satisfies the wave equation, and its axial factor obeysHere $c$ is the sound speed. Real $q_n$ permits propagation; positive imaginary $q_n$ produces an evanescent wave. A prescribed wall velocity proportional to one transverse factor excites that mode alone in the linear straight-channel model. The physical field is the real part of the complex expression.

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.

Let a homogeneous plate $0<x<L$ begin at uniform temperature $T_0$, cool with zero-temperature Dirichlet boundary conditions until time $t_b$, then switch to insulated Neumann boundary conditions. For thermal diffusivity $\kappa >0$, the Fourier sine series during cooling is $T=(4T_0/\pi ){\sum }_{n\ge 1,n\text{ odd}}\sin (n\pi x/L)e^{-\kappa (n\pi /L{)}^{2}t}/n$. Integrating the insulated heat equation shows that its integral average is conserved. All nonconstant Fourier cosine series modes then decay, proving that the final uniform temperature is $T_{\infty }=(8T_0/{\pi }^2){\sum }_{n\ge 1,n\text{ odd}}{e}^{-\kappa (n\pi /L{)}^2{t}_b}/n^2$. Its limits are $T_0$ at zero bath time and zero at infinite bath time. Instantaneous changes of boundary condition can create initial-boundary corner incompatibilities without invalidating the positive-time solution.

For real smooth coefficients with $p,w>0$, the Rayleigh quotient $\Lambda [y]=\int [p(y^{\prime }{)}^2+q{y}^2]dx/\int w{y}^{2}dx$ has the displayed first variation. Its stationary points solve the Sturm-Liouville equation on an admissible class whose boundary variations make $[p{y}^{\prime }\eta {]}_a^b=0$. Homogeneous Dirichlet boundary conditions do this by setting $\eta =0$ at the ends; homogeneous Neumann boundary conditions do it through $y^{\prime }=0$. More general self-adjoint boundary conditions can require a boundary contribution to the quadratic form.

For an eigenfunction, the weaker identity $[py{y}^{\prime }{]}_a^b=0$ suffices to give $\lambda =\Lambda [y]$ by integration by parts, but it does not by itself imply the condition on every variation. An explicit counterexample uses $p=q=w=1$ on $[0,1]$ and $y=\cos \pi x+\sin \pi x$. This solves $-y^{\prime \prime }+y=(1+{\pi }^2)y$ and has $[y{y}^{\prime }{]}_0^1=0$. Define a nearby familyIt keeps $y_{\epsilon }(0)=1$, $y_{\epsilon }^{\prime }(0)=\pi$, $y_{\epsilon }(1)=-1+\epsilon$, and $y_{\epsilon }^{\prime }(1)=\pi /(-1+\epsilon )$, so $[y_{\epsilon }{y}_{\epsilon }^{\prime }{]}_0^1=0$ for small $\epsilon$. Nevertheless, $\eta ={\partial }_{\epsilon }{y}_{\epsilon }{|}_0$ has $\eta (0)=0$, $\eta (1)=1$, and ${\int }_0^1{y}^{2}dx=1$, giving $\delta \Lambda =2[y^{\prime }\eta {]}_0^1=-2\pi \ne 0$. Boundary-energy cancellation alone therefore cannot replace a specified variational domain.

For cell diffusion, self-produced chemotactic agent and logistic proliferation, Neumann cosine modes reduce linear stability of the uniform state to a two-by-two matrix indexed by $s={\pi }^2(n^2/L_1^2+m^2/L_2^2)$. Its trace is negative and its determinant is $D(s)=\alpha s^2+(1+\alpha \kappa -\beta \kappa \eta )s+\kappa$. Chemotaxis can destabilize a mode precisely when $D(s)<0$, despite stability of the spatially uniform reaction system.

For $n_t=D_n{n}_{xx}-\chi (n{a}_x{)}_x+n(1-n)$, $a_t=D_a{a}_{xx}+\alpha n-\beta a$, the equilibrium $(1,\alpha /\beta )$ is stable without transport. A spatial mode of wavenumber $k$ has negative trace and determinant $D_n{D}_a{k}^4+(D_a+\beta D_n-\alpha \chi )k^2+\beta$. A continuous unstable band exists exactly under the displayed inequality. On a finite interval with Neumann boundary conditions, an allowed discrete wavenumber must additionally fall in that band. The high-wavenumber positive term is diffusive smoothing; the negative chemotactic term drives aggregation.

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.

For $H=H_0^1(U)\times H^1(U)$, the coupled formsatisfiesThe Poincaré 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.

For a divergence-form differential operator $-\nabla \cdot K\nabla$, its conormal derivative is the flux factor appearing in integration by parts. It generally differs from the ordinary normal derivative ${\partial }_{n}u=\mathbf{n}\cdot \nabla u$ when $K$ is anisotropic. Decomposing the gradient along a smooth boundary into its normal and tangential components gives ${\partial }_{n,K}u=(\mathbf{n}\cdot K\mathbf{n}){\partial }_{n}u+(\mathbf{n}\cdot K\mathbf{t}){\partial }_{\mathbf{t}}u$ in two dimensions. Thus zero normal derivative does not in general remove the boundary term in the usual symmetric weak formulation. A zero conormal derivative does remove it and produces the natural homogeneous Neumann boundary condition for that energy.

For sufficiently regular coefficients and a function $u$, this boundary quantity is the outward flux appearing when a divergence-form principal operator is integrated by parts. Its zero condition is the natural Neumann boundary condition for the corresponding bilinear form. With only bounded coefficients and a Sobolev solution, the weak formulation can encode that natural condition without asserting that a pointwise boundary derivative exists.

Consider the top-right corner $(b,d)$ of a rectangle. Suppose the top Dirichlet boundary condition is $u(x,d)=g(x)$ and the right Neumann boundary condition, with outward normal in the positive $x$ direction, is $u_x(b,y)=q(y)$. A solution that is a continuously differentiable function up to the corner necessarily satisfies $g^{\prime }(b)=q(d)$: differentiating the top boundary trace in its tangential $x$ direction gives $u_x(b,d)=g^{\prime }(b)$, while the right-edge condition gives the same partial derivative as $q(d)$.

This is a necessary compatibility condition, not an existence theorem. If it fails, a solution can still satisfy the interior equation and the boundary conditions on open edges while failing to have a continuous first derivative at that corner. In particular, homogeneous right-edge Neumann boundary conditions require $g^{\prime }(b)=0$ for a solution continuously differentiable at the top-right corner. Boundary values may remain continuous even when this derivative requirement fails.

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.

On $0<x<\ell$, homogeneous Neumann boundary conditions give spatial Fourier modes $\cos (m\pi x/\ell )$, $m=0,1,\dots$, with squared wavenumbers $q_m$. Differentiating twice proves that their Laplacian eigenvalues are $-q_m$, and their endpoint derivatives vanish. A dispersion relation $\sigma (q)$ therefore predicts the fastest permitted growing mode by maximizing $\mathrm{Re}\sigma (q_m)$ over integers $m$, rather than over all real wavenumbers. A continuous unstable band is insufficient if it contains no $q_m$.

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.

Take eight congruent hexagonal tiles with three pairwise nonadjacent distinguished faces. On tile indices modulo eight let $\sigma (j)=1-j$, $t(j)=-j$, and use $u_1(j)=3j$ for one assembly and $u_2(j)=3j+4$ for the other. Pair faces by reflection and leave fixed faces, and the three remaining faces, with Neumann boundary conditions. If $R{e}_j=e_{j+1}$, the displayed matrix intertwines all three face actions: its circulant coefficients satisfy $c_r=c_{-r}=c_{3r+4}$. Its eigenvalues are $1+(-1{)}^k+4\cos (\pi k/4)$, which are nonzero for $0\le k\le 7$. The transplantation theorem gives equal spectra. The second gluing graph is bipartite, with parts $\{0,5,6,7\}$ and $\{1,2,3,4\}$, so it is orientable. The first has the odd cycle $1,3,6,2,7,1$ and is nonorientable. Both graphs are connected with eight vertices and ten edges, and their bordered surfaces retract up to homotopy to the graphs. Both have Euler characteristic $-2$ and fundamental groups free of rank three. Neither is simply connected.

For a compact smooth Riemannian surface with smooth boundary and a pure Dirichlet boundary condition or pure Neumann boundary condition, the constant coefficient in the heat trace isThe equality is the Gauss-Bonnet theorem. Thus such isospectral manifolds have the same Euler characteristic. A simply connected compact connected bordered surface is a disk with $\chi =1$, whereas a nonorientable surface with $c\ge 1$ crosscaps and $b\ge 1$ boundary components has $\chi =2-c-b\le 0$. These cannot be isospectral in this setting. Corners and mixed boundary conditions contribute additional heat coefficients, so the smooth pure-boundary-condition hypothesis is essential.

For the heat equation with homogeneous Neumann boundary conditions at both ends of a uniform grid, even reflection supplies ghost values for a centered finite difference method. A fourth-order centered second derivative needs two ghost values at each end, obtained with the displayed rule for $j=1,2$. For a sufficiently smooth compatible solution, differentiating the Neumann boundary conditions in time and using the heat equation makes every odd spatial derivative vanish at the endpoints. Its even extension therefore retains the smoothness required by the fourth-order stencil; arbitrary incompatible initial data need care at the initial boundary corners.

The discrete cosine modes diagonalize this reflected spatial operator, with eigenvalues $-4h^{-2}{\sin }^2(\theta /2)[1+{\sin }^2(\theta /2)/3]$ and $\theta =j\pi /N$. Its most negative eigenvalue is $-16/(3h^2)$, so the forward Euler method is stable in the corresponding trapezoid-weighted norm when $\Delta t/h^2\le 3/8$. The Crank-Nicolson diffusion scheme is stable for every time step for this closure, but its amplification factor approaches $-1$ on highly damped spatial modes, allowing sign-alternating numerical transients. The ghost values are defined here as auxiliary samples outside the physical interval, rather than additional unknown initial data.

In the upper half-plane, the displayed harmonic function has Dirichlet boundary condition value $x^{-1/4}$ on the positive axis, zero Neumann boundary condition value on the negative axis, and decay at infinity. Its negative trace is $\sqrt{2}|x{|}^{-1/4}$, but its positive normal derivative is $x^{-5/4}/4$. The latter requires a Hadamard finite-part integral in a Fourier transform. For the $e^{ikx}$ convention the transformed traces are $\sqrt{2}\Gamma (3/4)(ik{)}^{-3/4}$ below and $-\Gamma (3/4)(-ik{)}^{1/4}$ above. The homogeneous corner mode $r^{-1/2}\sin (\theta /2)$ produces the additional lower integral transform proportional to $k^{-1/2}$. Higher singular corner modes require a larger distribution class than a locally integrable negative trace.

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.