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For $u_t=u_{xx}-a(t)u$, the displayed substitution converts the equation exactly into the heat equation $v_t=v_{xx}$. The scalar prefactor preserves homogeneous Dirichlet boundary conditions. Consequently each spatial eigenfunction $X_n$ with $X_n^{\prime \prime }=-{\lambda }_n{X}_n$ acquires temporal factor $\exp [-{\lambda }_{n}t-{\int }_0^{t}a(s)ds]$. The same algebra works with an evolution variable named $y$; calling that variable spatial does not alter the separation of variables. For $a(t)=1+\sin t$, the prefactor normalized to one at zero is $e^{-t+\cos t-1}$.

For dependence $e^{i(kz-\omega t)}$, a Helmholtz equation $({\Delta }_{\perp }+{\gamma }^2)\psi =0$ generates transverse-magnetic fields $e_{\perp }={\nabla }_{\perp }\psi$, $e_z=-i{\gamma }^2\psi /k$, $b_{\perp }=\omega \hat{z}\times {\nabla }_{\perp }\psi /(k{c}^2)$, $b_z=0$, for $k\ne 0$. The conducting wall requires a Dirichlet boundary condition on $\psi$. A transverse-electric representation is $e_{\perp }=\hat{z}\times {\nabla }_{\perp }\psi$, $e_z=0$, $b_{\perp }=-k{\nabla }_{\perp }\psi /\omega$, $b_z=i{\gamma }^2\psi /\omega$, with a Neumann boundary condition. These are families of modes, not a completeness assertion for multiply connected guides and their possible TEM fields.

For normal plane-wave incidence, the second-order Dirichlet boundary condition gives ${\psi }_s^{[2]}=-h{\partial }_z{\psi }_s^{[1]}$ at the mean plane, since ${\partial }_z^2{\psi }^{[0]}$ vanishes there. The Dirichlet-to-Neumann map for a Helmholtz half-space then yields $2kh\mathcal{B}h$. This nonlocal expression is not generally $2k^2{h}^2$; that local limit needs roughness varying slowly compared with the wavelength.

For a globally Lipschitz nonlinearity of constant $ϵ$, the map taking $v$ to the zero-Dirichlet Poisson solution with load $f(v)$ is a contraction in the gradient norm if the displayed inequality holds. Poincaré gives the factor $C_P$ from the Sobolev input to $L^2$, and testing the linear Poisson equation gives another factor $C_P$ back to the gradient norm. The condition yields a unique weak solution without requiring $f(0)=0$. With the optimal Poincaré constant it reads $ϵ<{\lambda }_1$.

Take seven copies of a scalene triangle, with sides labelled $a,b,c$ opposite its three vertices, and reflect adjacent copies across their glued side. For the first planar domain, the glued pairs are $a:(2,3),(6,7)$, $b:(4,6),(5,7)$, $c:(1,5),(3,7)$; for the second they are $a:(1,3),(5,7)$, $b:(2,6),(3,7)$, $c:(4,5),(6,7)$. Both coloured trees have central tile $7$. For triangles sufficiently close to equilateral the interiors do not overlap. If its three angles are distinct and between $\pi /4$ and $\pi /2$, the only reentrant vertices are the original central vertices, with angles four times the corresponding triangle angles. They force any proposed isometry to preserve the labelled central triangle, which is incompatible with the different attachments on its arms.

For the spectral proof index the nonzero vectors and covectors of ${\mathbb{F}}_2^3$ by $1,\dots ,7$, using binary coordinates. The Fano plane incidence matrix is $B_{y,x}=1_{y\cdot x=0}$. It satisfies $B^{T}B=2I+J$, so is invertible, and intertwines the point and line permutations of each of the three elementary transvections. Put $D=\mathrm{diag}(1,1,-1,1,-1,-1,1)$. The signed reflection matrices for the Dirichlet boundary condition are $-D{P}_{s}D$ and $-D{Q}_{s}D$, hence $C=DBD$ intertwines them. Its polar orthogonal factor gives a unitary equivalence of the Dirichlet Laplacians, preserving both the $L^2$ inner product and the Dirichlet energy of the piecewise functions. This produces a three-parameter family of noncongruent isospectral planar domains.

On $(0,L)$ with zero Dirichlet boundary conditions, the normalized trial $\sqrt{2/L}\sin (\pi x/L)$ has kinetic Rayleigh quotient ${\pi }^2/L^2$ and mean position $L/2$ by reflection symmetry. The Rayleigh-Ritz variational principle therefore gives the displayed upper bound for a real linear potential. For $ϵ\ge 0$ the lower bound ${\lambda }_1\ge {\pi }^2/L^2$ follows from the Poincaré inequality. For $ϵ\ne 0$ the trial is not an exact eigenfunction, so the upper bound is strict.

For $0<x<1$, consider the self-adjoint differential operator $L=(pd/dx{)}^{\prime }+q$ with $p=e^{-x}/(1-x)$ and $q=-e^{-x}/(1-x{)}^2$. The homogeneous solutions $u=x$ and $v=e^x-ex$ satisfy the left and right zero Dirichlet boundary conditions. Their Wronskian is $(x-1)e^x$, and $pW=-1$. Thus the displayed Dirichlet Green function is continuous, zero at both endpoints, and has unit flux jump $p(s)[G_x(s^{+},s)-G_x(s^{-},s)]=1$.

The right solution has a double zero, $v(s)\sim e(1-s{)}^2/2$. Consequently $G(x,s)F(s)$ is integrable at $s=1$ for fixed interior $x$ when $F(s)=-e^{-s}/(1-s{)}^2$, even though $F$ itself is not integrable there. The endpoint zero of the Wronskian causes no loss of this construction because the weighted Wronskian $pW$ stays nonzero. This illustrates the need to retain the integrating factor when constructing a Green function at a singular endpoint.

A skew-coupled elliptic system has zeroth-order couplings with opposite signs. For the displayed example with homogeneous Dirichlet boundary conditions, its weak formulation on $V=H_0^1(\Omega )\times H_0^1(\Omega )$ has the bilinear formThe coupling cancels for equal arguments, so $b((p,q),(p,q))=\Vert \nabla p{\Vert }_0^2+\Vert \nabla q{\Vert }_0^2$. On a domain satisfying the Poincaré inequality, this is a coercive bilinear form. With the product first-order Sobolev norm, the Cauchy-Schwarz inequality applied to $(\nabla p,\nabla q,-q,p)$ and $(\nabla v,\nabla z,v,z)$ gives $|b(P,T)|\le \Vert P{\Vert }_V\Vert T{\Vert }_V$. The Lax-Milgram theorem therefore applies even though the bilinear form is not symmetric. Eliminating $w=-\Delta u$ recovers ${\Delta }^{2}u+u=f$, involving the biharmonic operator.

Suppose the inverse $S:C^{0,\alpha }\to C_0^{2,\alpha }$ of a linear elliptic boundary value problem is bounded by $K$, and a nonlinear map satisfiesThen $T(v)=S(Q(v)+f)$ maps a sufficiently small closed ball into itself and is a contraction mapping when $\delta$ and $\Vert f{\Vert }_{C^{0,\alpha }}$ are sufficiently small. The Banach fixed-point theorem gives a solution of $Lu=Q(u)+f$ with zero Dirichlet boundary condition, unique within that ball. Invertibility follows from the Fredholm alternative for an elliptic Dirichlet problem when the homogeneous null space is zero.

For a time-independent strictly positive Dirichlet realization of an elliptic operator and $\psi \in L^2(U)$, the eigenfunction expansionsolves $u_t+Lu=0$ with homogeneous Dirichlet boundary conditions. For $t\ge \epsilon >0$, every power ${\lambda }_m^N$ times the exponential is bounded, so the series and all its time derivatives converge in all the Sobolev domains of powers of an elliptic Dirichlet operator. Elliptic regularity and the Sobolev embedding theorem give smoothness up to the spatial boundary for positive time. Parseval identity and the dominated convergence theorem give $u(t)\to \psi$ in $L^2$ as $t\downarrow 0$.

For incident amplitude one, ${\psi }_i=e^{i{p}_{i}x-i{q}_{i}z}$ with $q_i>0$, Taylor expansion of a small-height Dirichlet boundary condition gives ${\psi }_{\mathrm{sc}}^{[1]}(x,0)=2i{q}_{i}f(x)e^{i{p}_{i}x}$. If the mean height is constant, its mean transform is supported only at tangential wavenumber $p_i$, giving the displayed specular mean through first order. Small height alone is insufficient: deterministic mean height $a\cos Kx$ produces mean sidebands $p_i\pm K$. Stationary roughness has constant mean and therefore satisfies the first-order requirement.

For $n_t=D\Delta n+\gamma n$ in a sphere of radius $a$, with zero surface density and regularity at the origin, the lowest spherically symmetric normal mode is $n(r,t)=n_c\sin (\pi r/a)/(\pi r/a)\exp [(\gamma -{\pi }^{2}D/a^2)t]$. Substitution of $R=f/r$ in the radial Laplacian gives $f^{\prime \prime }+k^{2}f=0$, and regularity removes the cosine solution. The Dirichlet boundary condition gives $k=m\pi /a$; a radial mode with no interior zeros requires $m=1$. Thus diffusion leakage suppresses growth below the displayed threshold, while the mode is stationary at equality. The expression at $r=0$ uses its continuous limit. In a linear neutron multiplication model this is the competition between bulk production and surface loss.

On a shell $0<r_{-}<r<r_{+}$ with homogeneous Dirichlet boundary conditions, put $L=r_{+}-r_{-}$ and $u=r\psi$. The radial heat equation in three dimensions becomes $u_t=\kappa u_{rr}$, with zero values at both endpoints. Expanding $u(0,r)=r\psi (0,r)$ in a Fourier sine series gives the displayed solution, with $b_n=(2/L){\int }_{r_{-}}^{r_{+}}r\psi (0,r)\sin [n\pi (r-r_{-})/L]dr$. Each radial mode decays at the square of its wave number. The positive inner radius makes division by $r$ harmless throughout the shell.

Let $R_{jn}={\int }_{\partial \Omega }f{\partial }_n{v}_{jn}ds$ for a square sine adjoint test, and put ${\rho }_{jn}=e^{-{\omega }_n}{R}_{jn}$, ${\delta }_n=e^{-2{\omega }_n}$. The opposite-side normal-trace coefficients obeywith the same formulas for $L,R$. When the known Dirichlet boundary condition is integrated exactly, these are exact coefficients, since orthogonality eliminates all other unknown modes.

For the periodic-grid recurrencethe exact von Neumann stability analysis range at fixed $\mu >0$ is $\mu \ge 3$. The Fourier symbol gives $A(\theta ){\xi }^2-4\xi +1=0$, $A=3e^{2i\theta }+8\mu {\sin }^2(\theta /2)$. The root near one obeys $|\xi {|}^2=1+(6-2\mu ){\theta }^2+O({\theta }^4)$, proving instability for $\mu <3$.

For $\mu \ge 3$, $\mathrm{Re}A-3=6(\cos \theta -1{)}^2+4(\mu -3)(1-\cos \theta )\ge 0$. Writing the modal recurrence as $3v^{n+2}-4v^{n+1}+v^n=-(A-3)v^{n+2}$, the BDF2 discrete energy identity proves nonincrease of $|v^{n+1}{|}^2+|2v^{n+1}-v^n{|}^2$. Summing modes proves stability uniformly even when $\mu$ varies within $[3,\infty )$.

This shifted scheme is not a consistent standard heat discretization under $k=\mu h^2$: dividing its extra spatial-shift defect by $2k/3$ produces $3h{u}_x/k+O(h^2/k)$. On a finite interval with Dirichlet boundary conditions it also requires an extra boundary closure. Thus the periodic stability theorem is not a theorem of convergence to the heat equation or of stability for an unspecified boundary treatment.

Impose zero Dirichlet boundary conditions on an interval. The isoperimetric constraint $I[y]=1$ is regular because $D{I}_y[y]=2I[y]=2\ne 0$. The Lagrange multiplier rule applied to $J-\lambda (I-1)$ gives the Sturm-Liouville equationMultiplying by $y$ and using integration by parts gives $J[y]=\lambda I[y]=\lambda$. Thus the multiplier is the stationary value of the original functional, or its Rayleigh quotient. This calculation does not require $w$ to have one sign; positive-weight spectral conclusions require additional hypotheses.

If two stationary functions have different multipliers, multiplying their equations crosswise and subtracting gives $({\lambda }_n-{\lambda }_m)\int w{y}_n{y}_m=0$. The endpoint terms vanish. This proves the weighted orthogonality of Sturm-Liouville eigenfunctions. Higher stationary values need not be local minima of the constrained functional.

For a nonzero horizontal Fourier mode $k$, solve $(D^2-k^2{)}^{2}w=R{k}^2\theta$, $w=w^{\prime \prime }=0$, to obtain $w={\mathcal{W}}_R\theta$. Under homogeneous temperature Dirichlet boundary conditions, the sine modes diagonalize this self-adjoint operator, with eigenvalues $R{k}^2/(k^2+n^2{\pi }^2{)}^2$. The temperature linear operator is ${\mathcal{L}}_R=D^2-k^2+{\mathcal{W}}_R$. At a marginal mode $g=\sin (n\pi z)$ it has ${\mathcal{W}}_{R}g=(k^2+n^2{\pi }^2)g$.

If real matrices satisfy $A^T=A\le 0$ and $B^T=-B$, then $e^{tA}$ is a contraction and $e^{tB}$ is an orthogonal matrix for $t\ge 0$. Hence $\Vert e^{kA/2}{e}^{kB}{e}^{kA/2}{\Vert }_2\le 1$ for every $k\ge 0$. The bound is independent of commutativity and gives unconditional stability for a centered convection-diffusion semidiscretization with homogeneous Dirichlet boundary conditions. It does not itself bound the mesh-dependent commutators that enter a time-error estimate.

For a compact string embedding map coordinate, T-duality preserves its left-moving part and reverses its right-moving part. Consequently ${\partial }_{\tau }\tilde{X}={\partial }_{\sigma }X$ and ${\partial }_{\sigma }\tilde{X}={\partial }_{\tau }X$. A Neumann boundary condition becomes a Dirichlet boundary condition and conversely. A D-brane wrapping the circle loses one spatial worldvolume dimension; a D-brane transverse to it gains one in the dual description.

Consider a connected assembly of $n$ closed discs formed by identifying $e$ pairs of boundary arcs, each arc used once and all arcs in a given tile having disjoint endpoints. Its gluing graph has one vertex per tile and one edge per paired arc. A spanning tree joins the initial discs into a single closed disc; each of the remaining $e-n+1$ pairings attaches a band. This is a thickening of the gluing graph, so it is homotopy equivalent to that graph. Thus its Euler characteristic is $n-e$ and its fundamental group is the free group $F_{e-n+1}$. If the gluing graph is a tree, the assembled surface is itself a closed disc.

Suppose each pairing preserves the positive boundary parameter of the reference discs. A global orientation must then assign opposite orientation signs to adjacent tiles, because their induced boundary orientations on an interior seam must oppose one another. Such signs exist exactly when the gluing graph is bipartite. An odd gluing cycle therefore proves that the assembly is a nonorientable surface. A Dirichlet boundary condition or a Neumann boundary condition on an unpaired arc has no effect on this topological calculation.