a boundary-integral equation
Solving a PDE inside a region usually means finding the unknown everywhere in the interior — a lot of unknowns filling a volume. Boundary-integral equations turn that around with a clever observation: for a homogeneous equation like Laplace's, the solution inside is completely determined by data on the boundary, so why not reformulate the whole problem as an equation living only on the boundary surface? You drop one dimension — solve on the skin instead of the body — and recover the interior afterward by an integral.
The tools are layer potentials, built from the fundamental solution E. A single-layer potential spreads a density sigma over the boundary like a sheet of charge: u(x) = integral over boundary of E(x - y) sigma(y) dS(y). A double-layer potential spreads a density mu like a sheet of dipoles, using the normal derivative of E: u(x) = integral over boundary of (dE/dn_y)(x - y) mu(y) dS(y). Each automatically solves Laplace's equation inside (it is a superposition of fundamental solutions sourced on the boundary, hence harmonic in the interior). To match a Dirichlet or Neumann boundary condition you let x approach the boundary and demand the potential equal the given data. Because layer potentials have specific jump relations as you cross the boundary, this produces an integral equation for the unknown density sigma or mu over the surface — a Fredholm integral equation of the second kind, whose solvability is governed by the Fredholm alternative.
Why this is attractive: dropping a dimension is a huge saving, and for exterior problems (flow around an obstacle, scattering off a body) the boundary is finite even though the domain is infinite, so the method handles unbounded regions naturally with the correct decay built into E. This is the analytic foundation of the boundary-element method in engineering. Honest caveats: the resulting integral equations have singular kernels needing care; the jump relations and which layer to use depend on the boundary condition; and the choice between single-layer, double-layer, or a combination matters for whether the equation is uniquely solvable at every frequency (a real issue for the Helmholtz equation).
To solve the exterior Dirichlet problem for Laplace's equation around an obstacle, write u as a double-layer potential with unknown density mu. Letting x approach the boundary gives a second-kind integral equation (-(1/2) mu + integral of (dE/dn) mu = g) for mu on the surface alone.
A 3D problem becomes a 2D integral equation on the surface.
Layer potentials have jump discontinuities across the boundary and singular kernels, so the reduction needs care; the wrong choice of layer can make the integral equation non-uniquely solvable at certain frequencies (notably for Helmholtz).