From empty space to a source: meet Poisson
Across this whole rung you have studied Laplace's equation Laplacian u = 0 and its solutions, the harmonic functions — the steady states, the averagers, the smoothest fields a region can hold. But every one of those models described a region with nothing in it: no charge, no mass, no heater. A harmonic function is the potential of empty space. The honest next question is the one any physicist asks immediately: what if there is something in there — a lump of charge, a distribution of mass, a source dumping heat? The answer is the single most important inhomogeneous equation in all of analysis.
Poisson's equation is simply Laplace's equation with a source on the right: Laplacian u = f, or with the physics sign convention -Laplacian u = f. Here f is given — the density of mass, the density of charge, the rate of heat injection at each point — and u is the unknown potential it generates. Where f is zero the equation collapses straight back to Laplace's, so u is harmonic in every empty region; the source f is exactly the place where harmonicity is allowed to break. Poisson's equation is the precise mathematical statement of "this field is in equilibrium everywhere except where its sources sit."
The one source that matters: a point
Because Poisson's equation is linear, you can solve it the way you solved everything linear in this ladder: find the response to the simplest possible source, then add up. The simplest source is a single point — a unit of mass or charge squeezed onto one spot, an idealized spike written as the delta "function" delta(x). The response to that point source is the celebrated fundamental solution of the Laplacian, the field of one isolated point charge sitting at the origin. Everything else in this guide is built from it.
In three dimensions this field is the Newtonian potential of a point: it falls off like 1/r, exactly the gravitational potential of a point mass and the electrostatic potential of a point charge that Newton and Coulomb wrote down by experiment. The PDE confirms their inverse-square force law from pure symmetry: away from the origin the field is radial and harmonic, and the only radial harmonic function in 3D that decays is a constant over r. In two dimensions the answer is different and worth knowing — there is no 1/r solution; the radial harmonic function is log r, so the logarithmic potential -log r (up to constants) is the fundamental solution of the plane.
Fundamental solution Phi of -Laplacian Phi = delta (point source at origin): 1 dimension : Phi(x) = -|x| / 2 (grows linearly) 2 dimensions: Phi(x) = -(1 / 2pi) log r (logarithmic potential) 3 dimensions: Phi(x) = 1 / (4pi r) (Newtonian potential, ~ 1/r) where r = |x| = distance to the source Phi is harmonic everywhere EXCEPT at r = 0, where the source lives
Adding up sources: convolution solves Poisson
Now the payoff. A general source f is just a continuous cloud of point sources, one at every location y with strength f(y). Each contributes its own scaled, shifted fundamental solution Phi(x - y) f(y), and by the superposition principle the total potential is the sum — really an integral — of all of them. That integral is exactly the Newtonian potential of the whole distribution f, and it solves Poisson's equation on all of space: u(x) = integral of Phi(x - y) f(y) dy. You have just solved -Laplacian u = f in one clean stroke, with no series and no separation of variables.
This is a convolution, written u = Phi * f, and it is the elliptic cousin of the Gaussian convolution you used for the heat equation: in both cases the fundamental solution is the response to a point spike, and convolving it against the data spreads that response over every point of the source. The picture is concrete: stand at a point x, look out at every chunk of source around you, weight each by how strongly the 1/r (or log r) potential reaches you from there, and add. Faraway mass contributes little, nearby mass contributes a lot, and the singular tip of Phi is gentle enough that the integral converges even right inside the source.
It pays to be honest about why the singularity does not wreck the integral. Differentiating under the integral sign twice formally gives Laplacian u = integral of Laplacian Phi(x - y) f(y) dy = integral of delta(x - y) f(y) dy = f(x) — but Laplacian Phi is not an ordinary function, it is the delta, so that line is a calculation in the language of distributions, not a naive swap of derivative and integral. The careful version is a short argument using Green's identities to isolate the singular point. The slogan to keep: the Laplacian of the fundamental solution is a point spike, which is the whole reason convolving with it inverts the Laplacian.
From all of space to a box: Green's functions
The convolution formula solves Poisson on all of space, where the only condition is decay at infinity. But the real problems of this rung live in a bounded region with prescribed Dirichlet or Neumann data on the wall. The Newtonian potential alone does not respect those walls: it solves -Laplacian u = f correctly, but it leaves the wrong values on the boundary. The fix is the same trick as a boundary corrector — add a harmonic function that touches up the boundary without disturbing the source.
Stitching the free-space fundamental solution together with that boundary-fixing harmonic correction gives the region's Green's function G(x, y): the potential at x of a point source at y that also vanishes on the boundary. With it in hand, the full Dirichlet problem -Laplacian u = f with u = g on the wall has a single tidy formula — one integral of G against the source f over the interior, plus a boundary integral of the normal derivative of G against the data g. The Poisson integral formula you met in the previous guide is precisely this construction carried out for the disk, where the boundary term is everything because the source is zero.
What the source does to the field, and honest fine print
Now read off what Poisson buys you physically. Recall that a harmonic function equals its own average over every small sphere — the mean-value property. Poisson's equation is the exact correction to that: where f is positive, u dips below its spherical average; where f is negative, u rises above it. The source bends the field away from flat the way a load bends a stretched membrane downward beneath the weight. This is why the maximum principle is strictly a harmonic statement: a genuine source can manufacture an interior peak or trough that a sourceless field never could.
A beautiful structural fact closes the circle with the Dirichlet problem. The solution of -Laplacian u = f is the field that minimizes the energy integral of (1/2)|grad u|^2 - f u — stretching energy traded against the work the source does. Setting f = 0 recovers Dirichlet's principle from the earlier guide: with no source, the harmonic function is simply the least-stretched field fitting the boundary. Poisson's equation is Laplace's principle of least effort, with a source paying the field to deform.
Two honest cautions before you go. First, the convolution formula assumes f decays fast enough; a source that does not die off at infinity (an infinite uniform charge) makes the integral diverge, and you must work locally instead — the global formula is a convenience, not a law. Second, even a rough, merely-integrable source f produces a remarkably smooth potential u: the potential is always two derivatives better behaved than its source, a precise statement called elliptic regularity. This smoothing is real but bounded — do not over-promise it; a discontinuous source gives a u whose second derivatives can still be discontinuous, just no worse than f itself. As always in this subject, the fundamental solution is the keystone: know it, and the whole arch of elliptic theory stands.