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An absorbing boundary removes a diffusion when it is reached. The forward density and backward observables satisfy Dirichlet boundary conditions there.

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 $u_{tt}=\Delta u+\alpha u$ with a homogeneous Dirichlet boundary condition, let ${\lambda }_1>0$ be the first Dirichlet Laplacian eigenvalue. On a bounded one-dimensional interval, finite-energy displacement remains bounded for all time and all initial data exactly when $\alpha <{\lambda }_1$. The conserved energy $\frac{1}{2}(\Vert u_t{\Vert }_2^2+\Vert \nabla u{\Vert }_2^2-\alpha \Vert u{\Vert }_2^2)$ controls the gradient and, by the Poincaré inequality and one-dimensional integration, the maximum norm. At equality a mode can grow linearly; above it a mode can grow exponentially. Bounded oscillations need not have a long-time limit.

For an annulus $a<r<1$ with $0<a<1$, these factors interpolate the two boundary coefficients of a Fourier harmonic of positive integer order $n$ in a harmonic function. They obey $H_n(1)=1$, $H_n(a)=0$, $K_n(1)=0$, $K_n(a)=1$. Thus a mode with outer coefficient $v$ and inner coefficient $w$ has radial factor $v{H}_n+w{K}_n$.

To derive them, separation of variables in Laplace's equation in polar coordinates gives $r^2{R}^{\prime \prime }+r{R}^{\prime }-n^{2}R=0$, whose solutions are $r^n,r^{-n}$. Imposing the two Dirichlet boundary conditions on $R=\alpha r^n+\beta r^{-n}$ gives $\alpha +\beta =v$ and $a^n\alpha +a^{-n}\beta =w$. Solving yields $\alpha =(v-a^{n}w)/(1-a^{2n})$ and $\beta =a^n(w-a^{n}v)/(1-a^{2n})$, proving the formula. For the zero angular mode, the analogous factor is $v+(w-v)\log r/\log a$, since its radial solutions are $1$ and the logarithm $\log r$.

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.

Let $Ay=(p{y}^{\prime }{)}^{\prime }+qy$ have real eigenfunctions $u_n$ obeying $A{u}_n={\lambda }_{n}w{u}_n$ and homogeneous Dirichlet boundary conditions, with $N_n={\int }_a^{b}w{u}_n^2>0$. Suppose $Ay+\mu wy=f$, with possibly nonzero endpoint values. The Lagrange identity for a Sturm-Liouville operator givesDivision by $({\lambda }_n+\mu )N_n$ gives the displayed coefficient whenever the denominator is nonzero. At resonance in a differential equation, equality of the forcing projection and the boundary term is the Fredholm solvability condition. The Sturm-Liouville eigenfunction expansion represents the interior solution in the weighted inner product sense; its zero endpoint terms do not reproduce nonzero boundary traces by pointwise substitution. A smooth lift of the endpoint data reduces the calculation to homogeneous Dirichlet boundary conditions.

Subtract a known function with the required boundary trace so that the remaining unknown has homogeneous boundary conditions. For an interval with Dirichlet boundary conditions $g(t),h(t)$, $\ell =(1-x/L)g+(x/L)h$ is a simple lifting. In $w_t=w_{xx}-cw$, $z=w-\ell$ has forcing $-{\ell }_t-c\ell$ and zero endpoint values. This improves Fourier sine series convergence.

The value assigned to or approached by a function at a point of the boundary of its domain. Prescribing a boundary value is a Dirichlet boundary condition.

Two bounded harmonic functions continuous to the quadrant edges with the same Dirichlet boundary conditions are equal. Their difference is bounded and zero on both axes. Odd reflection across each axis gives a bounded harmonic extension away from the corner; the removable singularity for a bounded harmonic function fills in that corner. The harmonic Liouville theorem makes that extension constant, and oddness makes the constant zero. This does not exclude unbounded examples such as the zero-boundary harmonic growth in a quadrant.

For the exterior of the unit disc in two dimensions, with the sign convention ${\Delta }_{x}G=+\delta (x-y)$, this is the Dirichlet Green function that vanishes on the circle and is bounded at infinity. The method of images places an opposite logarithmic source at the inverse point $y/|y{|}^2$. The constant $-(2\pi {)}^{-1}\log |y|$ enforces the zero boundary value. The inversion in a circle identity $|x-y|=|y||x-y/|y{|}^2|$ for $|x|=1$ verifies that condition directly.

The outward normal of the exterior domain is $-x$ on the unit circle. Its normal derivative gives the positive exterior Poisson kernelFor bounded harmonic functions this kernel integrates boundary data to the interior value. Boundedness at infinity is essential to uniqueness: without it, a multiple of $\log |y|$ can be added without changing the circle's Dirichlet boundary condition.

The capacitor gap aspect ratio compares the gap $d$ with a lateral plate width $W$. For rectangular plates of widths $W_x,W_z$, require both $d/W_x\ll 1$ and $d/W_z\ll 1$ for the leading parallel-plate capacitor approximation. Large area alone is insufficient: a long narrow strip can have large area and still have $d$ comparable with its smaller width. In addition, the observation point must be many gap widths from an edge. Reducing $d/W$ improves the interior approximation without making the edge itself disappear.

In a two-dimensional straight gap, write the electric potential as the infinite-plate solution plus a perturbation:The perturbation satisfies the Laplace equation and vanishes on the two ideal plates. Separation of variables in a semi-infinite strip, with $s$ the distance inward from its entrance, gives transverse modesThus edge disturbances have a natural penetration scale set by $d$. Modes forbidden by a particular symmetry have zero coefficients; this expression is an explanation of localization in an ideal strip, not a universal numerical correction to finite-plate capacitance. The outside boundary conditions determine the coefficients.

There is no single aspect-ratio cutoff guaranteeing an assigned accuracy: the requested observable, plate shape and thickness, surroundings, dielectric distribution, and guard geometry all matter. The small-gap approximation is local and is not uniform across an edge. It also does not justify extrapolating $C\simeq ϵA/d$ to arbitrarily large $d$, where finite electrical conductors and their external electric field dominate. At fixed $V$, formally shrinking an ideal gap makes the interior electric field grow like $1/d$; real electric currents across an imperfectly insulating gap, surface roughness and contact eventually invalidate that ideal limit.

With homogeneous Dirichlet boundary conditions, centered differences turn one-dimensional diffusion into a symmetric negative-definite matrix and constant advection into a skew-symmetric matrix. Their sum dissipates the discrete Euclidean norm, giving mesh-uniform energy stability for every fixed advection coefficient.

For the centered second difference and centered first difference on a finite interval with zero Dirichlet boundary conditions, the diffusion matrix is symmetric negative definite and the drift matrix is skew-symmetric. Discrete summation by parts gives the displayed mesh-weighted identity. It proves contraction in the discrete L2 norm without needing periodic modes, positive stencil coefficients or simultaneous diagonalization.

A particle confined by a Dirichlet boundary condition to a disc of radius $a$ has angular functions $e^{im\varphi }$, $m\in \mathbb{Z}$, and regular radial Bessel functions of the first kind $J_{|m|}(kr)$. Their boundary zeros give the displayed energies. The ground state belongs to $m=0$: nonzero angular modes add a nonnegative centrifugal term to the kinetic-energy form. At the origin exclude the singular radial solution, rather than accepting every square-integrable logarithmic function as a physical eigenstate.

For a quadratic potential energy, shifting a path to its classical solution leaves an exactly quadratic fluctuation action with homogeneous Dirichlet boundary conditions. The first variation vanishes by the Euler-Lagrange equation. The Gaussian path integral prefactor is an inverse square root of a regulated functional determinant and does not depend on the endpoints. At conjugate times the ordinary formula must instead be read by distributional continuation.

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.

A coercive zero-Dirichlet weak form gives a bounded solution map from $L^2$ into $H_0^1$. Composing with the compact Sobolev embedding into $L^2$ gives the compact inverse $K$. Transposing the bilinear form produces its Hilbert-space adjoint inverse $K^{\ast }$. The Fredholm compatibility condition for a shifted equation is orthogonality of $f$ to the homogeneous adjoint solutions. Indeed $(Kf,w)=(f,w)/\lambda$ when $K^{\ast }w=w/\lambda$, and the zero spectral parameter is excluded at resonance by coercivity.

For a fixed uniformly elliptic operator with $C^{0,\mu }$ coefficients on a bounded smooth connected domain, nonnegative solutions with zero Dirichlet boundary condition satisfy the displayed estimate when ${\Omega }^{\prime }$ is nonempty and compactly contained in $\Omega$. If it failed, normalize the global supremum to one. The forcing and the interior infimum then tend to zero. The global Schauder estimate and compact embedding of Hölder spaces produce a nonzero homogeneous limit with an interior zero, contradicting the strong minimum principle for elliptic operators. An arbitrary bounded zeroth-order coefficient is allowed: replace $c$ by $\min (c,0)$ for the positivity argument. Keeping global boundary control prevents the normalized maximum from disappearing in the limit.

A conforming finite element space is a finite-dimensional vector subspace of the solution's variational space $V$. For a second-order problem with homogeneous Dirichlet boundary conditions, continuous piecewise-polynomial functions vanishing on the boundary form such a subspace of the zero-boundary Sobolev space $H_0^1(\Omega )$. Conformity allows exact Galerkin orthogonality and the Céa lemma.