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A fundamental solution of a linear PDE operator L is a distribution Φ satisfying LΦ = δ (the Dirac delta) in all of ℝⁿ, without boundary conditions. It represents the response to a point source in free space and is the building block from which Green's functions and general solutions are constructed. The fundamental solution for the Laplacian is the Newton potential (1/|x| in 3D), for the heat operator it is the Gaussian heat kernel, and for the wave operator it is a distribution supported on the light cone. These explicit formulas encode the essential character of each PDE type.
Fundamental solutions are the atoms of linear PDE theory. A fundamental solution Φ for an operator L satisfies LΦ = δ in the distributional sense, meaning it is the response to an idealized point source in all of space without boundaries. Unlike Green's functions, fundamental solutions are intrinsic to the operator itself and independent of any domain or boundary conditions. They exist for all constant-coefficient linear operators with nonzero principal symbol, as guaranteed by the Malgrange-Ehrenpreis theorem.
The fundamental solution of the Laplacian reflects the geometry of diffusion in different dimensions. In ℝ³, Φ(x) = 1/(4π|x|) is the Newtonian potential—the electrostatic potential of a point charge. In ℝ², Φ(x) = -(1/2π)ln|x| is the logarithmic potential, which grows without bound at infinity, reflecting the recurrence of two-dimensional random walks. In ℝⁿ for n ≥ 3, Φ(x) = c_n/|x|^(n-2). These formulas encode how influence from a point source decays with distance, and they underpin all of classical potential theory.
The heat kernel K(x,t) = (4πkt)-n/2exp(-|x|²/(4kt)) for t > 0 is the fundamental solution of the heat operator ∂_t - kΔ. It is a Gaussian whose width grows as √(kt), describing how a point source of heat spreads diffusively. Two crucial properties are immediate from the formula: the infinite speed of propagation (K > 0 for all x when t > 0) and the smoothing effect (K is infinitely differentiable for t > 0 despite the singular initial data). The heat kernel appears throughout mathematics—in probability (transition density of Brownian motion), geometry (local index theorem), and number theory (Jacobi theta function).
For the wave operator ∂²_t - c²Δ, the fundamental solution depends critically on the spatial dimension. In one dimension, it is the Heaviside function H(ct - |x|)/(2c), which is supported inside the full light cone—a pulse rings forever. In three dimensions, it is a distribution supported exactly on the light cone |x| = ct, giving sharp wave fronts (Huygens' principle). In two dimensions, it is supported inside the light cone, producing the familiar trailing waves seen when a stone is dropped in water. These dimension-dependent behaviors are among the most striking results in PDE theory and have profound physical consequences.