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 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 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 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.

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

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.

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 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.

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