Boundary condition

A boundary condition prescribes values of an unknown function or its derivatives at the boundary of the domain of a differential equation.

A boundary condition at spatial infinity specifies limiting behaviour as the spatial coordinate leaves every bounded set, such as ${\lim }_{x\to +\infty }u(x,t)=g(t)$ on a half-line. A pointwise limit for each time is different from uniform control over a time interval. Maximum-principle and uniqueness proofs on unbounded domains must use a suitable uniform boundedness, growth restriction or barrier hypothesis to justify passage from finite domains to infinity.

A linear boundary condition can relate traces at different endpoints rather than prescribe data independently at each endpoint. For a second-order differential equation, examples are $a_{1}y(a)+b_1{y}^{\prime }(a)+c_{1}y(b)+d_1{y}^{\prime }(b)=0$ together with another independent relation. The resulting operator domain determines whether the boundary form vanishes. Coupling by itself guarantees neither self-adjointness nor its failure.

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.

A dynamic boundary condition includes a time derivative of the boundary trace. For a half-line heat equation, $u_t(0,t)+\alpha u_x(0,t)+\beta u(0,t)=f(t)$ gives a separate boundary evolution coupled to the interior flux. Its Laplace transform contains the initial trace $u_0(0)$. The sign of $u_x$ differs from that of the outward normal derivative at a left endpoint.

For a smooth heat equation solution $w_t=\Delta w$ with $w_t+\alpha {\partial }_{n}w=0$ on the boundary and nonnegative $\alpha$, Green's first identity gives the displayed dissipation identity for $E={\int }_V|\nabla w{|}^2$. The outward normal derivative is used. Writing the boundary loss as $\alpha ({\partial }_{n}w{)}^2$ avoids division by $\alpha$ and remains valid where $\alpha =0$. A stationary solution is harmonic with $\alpha {\partial }_{n}w=0$; it need not be a particular fixed-data Dirichlet problem solution.

For the dynamic boundary condition for the heat equation with $\alpha =2a$, $\beta =a^2$, the boundary resolvent denominator is $(\sqrt{p}-a{)}^2$. With the principal square root, its inverse kernel isEquivalently, $J_a={\int }_0^{\infty }r{e}^{ar}P(x+r,t)dr$ using the Heat Poisson kernel. This representation fixes the repeated-root sign and remains valid for either sign of $a$.

For $a>0$, the heat equation on a half-line with $u_t+2a{u}_x+a^{2}u=0$ at the left endpoint admits $u=e^{-ax+a^{2}t}$. The solution is spatially decaying and temporally growing. The boundary resolvent has a double pole at $p=a^2$, allowing generalized contributions proportional to $t{e}^{a^{2}t}$. For $a<0$ the root $\sqrt{p}=a$ is absent on the principal square root branch.

A clamped endpoint of a fourth-order beam or eigenvalue problem imposes both zero displacement and zero slope, $y=y^{\prime }=0$. These conditions remove the boundary terms produced by integrating a fourth derivative by parts twice.

A periodic boundary condition identifies opposite ends of a domain by requiring the solution and the derivatives needed by the equation to agree there.

A twisted boundary condition identifies the ends of an interval up to a fixed unit-modulus complex number. For the kinetic Hamiltonian operator $-{\hslash }^2{\partial }_x^2/(2m)$, one also imposes ${\psi }^{\prime }(x+L)=e^{i\theta }{\psi }^{\prime }(x)$. In integration by parts, the endpoint terms then cancel, giving the usual self-adjoint ring realization. The allowed plane waves have momenta $\hslash (2\pi n+\theta )/L$.

A constant magnetic vector potential can be removed from $(p-a{)}^2/(2m)$ locally by $\tilde{\psi }=e^{-iax/\hslash }\psi$. If $\psi$ is periodic, $\tilde{\psi }$ has twist $\theta =-aL/\hslash$. Thus the Aharonov-Bohm effect survives a locally vanishing vector potential as a global condition on the wavefunction.

An antiperiodic boundary condition is a phase twist of $\pi$ across the boundary. A Fourier mode then has $k=2\pi (n+1/2)/L$. It occurs in the even-fermion sector of a periodic Jordan–Wigner transformation and in imaginary-time fermionic thermal fields, for different underlying reasons.

A vanishing boundary condition requires the relevant field, variation, or flux to tend to zero on the boundary. It removes boundary terms produced by integration by parts.

An initial condition prescribes the state of a time-dependent problem at a chosen initial time.

Initial velocity specifies the time derivative of the displacement field at the starting time. A second-order wave equation requires it independently of initial displacement. Both vanish for a causal scattered field before an incident pulse reaches its scatterer.

Initial displacement specifies the configuration of a wave equation at the starting time. For a second-order evolution equation it is paired with the initial velocity. Specifying only the boundary forcing does not determine these initial data.

A Dirichlet boundary condition prescribes the value of the unknown function on the boundary.

For the forward diffusion equation $u_t=\kappa u_{yy}$ on $0<y<L$, with $\kappa >0$ and homogeneous Dirichlet boundary conditions, a solution continuous on the closed initial-boundary domain can take initial data $u(0,y)=\varphi (y)$ only if $\varphi (0)=\varphi (L)=0$. These equalities express the same corner value through the initial condition and the boundary condition.

An incompatible endpoint value does not prevent an interior or L2 space initial-data solution. For example, on $0<y<1$ the initial ramp $\varphi (y)=y$ givesIt satisfies both zero endpoint values for every $t>0$ and tends to $y$ at every fixed $0<y<1$ as $t\downarrow 0$, also in the L2 norm. It cannot extend continuously to $(0,1)$ with both the initial value one and the boundary value zero. Each sine term vanishes at the endpoint, so the series cannot silently repair that pointwise contradiction.

If the equation is written $a{u}_t=u_{yy}$, the intended forward heat coefficient requires $a>0$. For $a<0$, the ramp's coefficients would grow like $e^{n^2{\pi }^{2}t/|a|}/n$, failing square summability for every $t>0$. This example distinguishes an incompatible corner from the separate ill-posedness of backward diffusion.

Dirichlet boundary data prescribe the boundary values of a solution as a function $\phi$ on the boundary of its domain. Continuity on a polygon requires agreement of the values at corners, but continuity alone does not ensure solvability if the equation loses uniform ellipticity at a boundary side.

In a fixed-temperature convection problem, the boundary temperature is prescribed, and its perturbation satisfies a thermal Dirichlet boundary condition. A homogeneous condition means the boundary follows the prescribed conductive background temperature. This does not necessarily mean a vertical wall has a spatially constant temperature; that additional assumption can conflict with a background vertical gradient.

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.

A tangential boundary derivative differentiates the boundary restriction of a function along a tangent vector. Prescribing it fixes the boundary value only up to a constant on each connected boundary component. This differs from a Neumann boundary condition, which specifies a normal derivative.

A Robin boundary condition prescribes a linear combination of a function and its outward normal derivative on the boundary.

For $-(p{u}^{\prime }{)}^{\prime }=f$, $u(b)=0$ and $u^{\prime }(c)+\kappa u(c)=0$, integration by parts gives the displayed endpoint coefficient. Ordinary derivative data and conormal data $p(c)u^{\prime }(c)+\kappa u(c)=0$ are different boundary conditions. If $p$ is bounded above and below by positive constants and $\kappa \ge 0$, the Poincaré inequality on functions vanishing at $b$ proves continuity and coercivity. The symmetric form defines an energy norm, and Galerkin orthogonality gives exact best approximation in that norm.

On a bounded Lipschitz domain, the Poisson equation $-\Delta u=f$ with Robin boundary condition ${\partial }_{n}u+\beta u=0$ has the displayed weak formulation in $H^1(\Omega )$ when $\beta$ is a bounded boundary coefficient. Boundary integrals use Sobolev traces. It follows from Green's first identity, requiring only first weak derivatives of $u$. If $\beta$ is uniformly positive, testing the difference of two solutions against itself proves uniqueness: its gradient and its boundary Sobolev trace vanish. The Poincare inequality with a boundary trace also makes this a coercive bilinear form.

On a bounded region, two sufficiently regular solutions of the Poisson equation with the same forcing and boundary condition $\alpha {\partial }_n\varphi +\varphi =\gamma$, where $\alpha \ge 0$, have harmonic difference $w$. Green's first identity gives $\int |\nabla w{|}^2={\int }_{\partial V}w{\partial }_{n}w=-{\int }_{\partial V}\alpha ({\partial }_{n}w{)}^2\le 0$. Hence $w$ is constant on each component, and the boundary condition makes that constant zero. The proof never divides by $\alpha$, so it includes variable coefficients and Dirichlet portions where $\alpha =0$.

For a decaying harmonic function on a semi-infinite strip with equal outward Robin boundary conditions and reflection-symmetric end data, the top and bottom value traces agree. The global relation and its reflected spectral companion express their transforms through $D(k)$. With $H(k)=\frac{1}{2}{\int }_0^{\ell }{e}^{ky}g(y)dy$, the relation is $\psi (-ik)+\psi (ik)=2H(k)/D(k)$. Real zeros of $D$ require compatible data; unrestricted negative Robin parameters do not guarantee a decaying solution.

Let ${\lambda }_n$ be the transverse eigenvalues of a Sturm-Liouville problem for the two Robin boundary conditions. For a harmonic strip solution decaying in the longitudinal direction, projection of the prescribed end Neumann boundary condition onto every nonpositive eigenvalue must vanish. Positive modes give $q_n(x)=-g_n{e}^{-\sqrt{{\lambda }_n}x}/\sqrt{{\lambda }_n}$. Zero modes are affine and negative modes are oscillatory, so neither can contribute to a decaying solution. The condition also establishes uniqueness within the decay class.

For $Lu=-{\partial }_j(a^{ij}{\partial }_{i}u)+b^i{\partial }_{i}u+cu$ with smooth coefficients on a bounded smooth domain, the weak formulation of $(L+\lambda )u=f$ and conormal Robin boundary condition $a^{ij}{\partial }_{i}u{\nu }_j+\beta u=0$ isThe bilinear form is bounded by the Sobolev trace theorem. If $a^{ij}$ satisfy the positive uniformly elliptic operator inequality, the multiplicative trace inequality on a bounded smooth domain and the Young inequality show that the form is coercive when $\lambda$ is sufficiently large, even if $\beta$ and $c$ have negative parts. The Lax-Milgram theorem then gives a unique weak solution for every $f\in L^2(U)$.

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.

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.

A far-field boundary condition prescribes the limiting behavior of a solution as one or more spatial coordinates tend to infinity.