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The biharmonic equation is ${\nabla }^{4}f=0$, where the biharmonic operator is the square of the Laplacian. In planar Stokes flow, taking the curl of the momentum equation and using a Cartesian streamfunction produces this equation. Its decaying spatial Fourier series modes have form $(A+By)e^{-|k|y}{e}^{ikx}$.

The biharmonic operator applies the Laplacian twice. Under clamped boundary conditions, two integrations by parts give the positive quadratic form

This is the closure of the compactly supported test functions $C_c^{\infty }(U)$ in the $H^2(U)$ norm. On a smooth bounded domain it consists precisely of the $H^2$ functions with zero value and zero normal derivative traces. Tangential first derivatives then vanish as well. The clamped Hessian identity makes the Laplacian norm equivalent to the full $H^2$ norm on this space, enabling a Lax-Milgram theorem treatment of the clamped biharmonic problem.

A differential operator maps a function to an expression depending on finitely many of its derivatives. A linear differential operator has the form $P={\sum }_{|\alpha |\le m}{a}_{\alpha }(x){\partial }^{\alpha }$; its order is the largest derivative order with nonzero coefficient. The Laplacian is a second-order example, and boundary conditions specify the domain when defining an operator on a function space.

Elliptic regularity says that solutions of an elliptic equation gain derivatives wherever the coefficients and forcing are regular. For the Laplacian, local estimates typically control two derivatives of $u$ by $\Delta u$ together with a lower-order norm of $u$.

The Fischer inner product on polynomials is $⟨x^{\alpha },x^{\beta }{⟩}_F=\alpha !{\delta }_{\alpha \beta }$. Multiplication by $x_i$ is adjoint to differentiation by $x_i$. Therefore multiplication by $|x{|}^2$ is adjoint to the ordinary Laplacian, yielding the harmonic decomposition of homogeneous polynomials by finite-dimensional orthogonality.

On a square mesh, the dimensionless five-point Laplacian is ${\Gamma }_{5}u=u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j}$. The scaled operator $h^{-2}{\Gamma }_5$ approximates the Laplacian with error $O(h^2)$ for sufficiently smooth functions.

Fourier space labels a field by the frequency or wavevector variables of its Fourier transform, rather than by its original coordinates. Spatial gradients act by multiplication by $i\mathbf{k}$, and the negative Laplacian acts by $k^2$.

If ${\mathcal{P}}_{\ell }$ is the space of degree-$\ell$ homogeneous polynomials and ${\mathcal{H}}_{\ell }$ its harmonic polynomials, then ${\mathcal{P}}_{\ell }={\mathcal{H}}_{\ell }\oplus |x{|}^2{\mathcal{P}}_{\ell -2}$. Under the Fischer inner product, the second summand is orthogonal to the kernel of the Laplacian. Iteration expresses every polynomial restricted to the unit sphere as a sum of spherical harmonics.

The healing length of a repulsive homogeneous condensate is the length at which density-gradient kinetic energy and interaction energy balance. The displayed convention sets the dimensionless Laplacian coefficient to one in the stationary Gross–Pitaevskii equation. A hard-wall number density profile has width $\sqrt{2}{\ell }_h$; definitions without the factor $\sqrt{2}$ occur, so the equation and convention must be specified.

A fixed sphere in a slowly varying incident Stokes flow requires an external force $\mathbf{F}=-6\pi \mu a{\mathbf{u}}_{\infty }$ and external couple $\mathbf{G}=-4\pi \mu a^3{\mathbf{\omega }}_{\infty }$ to leading order. These follow from Faxén translation law and Faxén rotation law; the incident Stokeslet supplies ${\mathbf{u}}_{\infty }$ and ${\mathbf{\omega }}_{\infty }$. Finite-radius Laplacian terms enter the translational force at the next nonzero order. The signs here refer to applied holding loads, not hydrodynamic loads.

For pressureless matter in Newtonian gauge in cosmology with a common metric potential $\Phi$, let $\theta =\nabla \cdot \mathbf{v}$. The linear equations ${\delta }_m^{\prime }=-\theta +3{\Phi }^{\prime }$ and ${\theta }^{\prime }=-\mathcal{H}\theta -{\nabla }^2\Phi$, where primes denote conformal time, yieldThis follows by differentiating the first equation and eliminating $\theta$. On deeply subhorizon scales with a slowly varying matter potential, the metric time derivatives are small compared with its spatial Laplacian.