Two different objects are called Poisson's formula

The name "Poisson's formula" denotes two different objects that should not be conflated. The two-dimensional wave formula of Poisson's formula in two dimensions by descent is the weighted disk integral $u(x,t)=\frac{1}{2\pi c}[\frac{\mathrm{\partial }}{\mathrm{\partial }t}{\int }_{B_{ct}(x)}\frac{u_0(y)dy}{\sqrt{c^2{t}^2-\mathrm{\mid }y-x{\mathrm{\mid }}^2}}+{\int }_{B_{ct}(x)}\frac{u_1(y)dy}{\sqrt{c^2{t}^2-\mathrm{\mid }y-x{\mathrm{\mid }}^2}}],$ whose kernel $(2\pi c{)}^{-1}(c^2{t}^2-\mathrm{\mid }y-x{\mathrm{\mid }}^2{)}^{-1\mathrm{/}2}$ is an interior singular weight on the expanding disk, while the harmonic Poisson kernel of Poisson kernel of a Euclidean ball integrates boundary data against a positive density on the sphere. The two solve different problems — an initial-value (Cauchy) problem in space-time versus the Dirichlet boundary-value problem — use respectively a time parameter $t$ and a fixed radius $R$. The shared name does not identify their kernels or transfer estimates between these problems.

The wave formula is stated for the speed-$c$ convention of Wave equation, Cauchy data and wave speed; the harmonic kernel is the ball boundary-value kernel of the Poisson-problem page. The remark asserts no new mathematics: it isolates the naming collision so that no consumer imports a boundary-value estimate into the wave representation.