Green's functions and Poisson kernels

For minus the Laplacian, a fundamental solution satisfies the distributional equation with a unit delta source in the whole space. In dimension at least three it is proportional to the distance raised to two minus the dimension; in two dimensions it is minus the logarithm divided by two pi. A Dirichlet Green function adds a harmonic correction so that its boundary values vanish. The correction depends on the domain, even though the local singularity near the source is inherited from the fundamental solution.

On the unit interval the boundary-adapted kernel is especially transparent. For fixed source location y, use one affine function to the left and another to the right, join them continuously, enforce zero at both endpoints, and arrange a derivative jump of minus one. The second distributional derivative then has a negative delta jump, so applying minus that derivative gives the required positive unit source. This direct sign check is safer than memorizing a representation with an unspecified Laplacian convention.

For continuous data on the unit circle, the disk Poisson formula integrates the boundary data against a positive normalized weight. At the center the weight is constant, so the value is the ordinary circle average. Away from the center it emphasizes nearby boundary angles while retaining total mass one. The resulting function is harmonic in the disk and extends continuously to the prescribed data on the circle. Positivity immediately keeps its values between the boundary minimum and maximum.

The disk formula separates angular frequencies: the boundary cosine mode of integer order m extends as radius to the m times cosine of m theta. High frequencies decay quickly away from the boundary, while the constant mode persists unchanged. This is spatial harmonic extension, not time-dependent heat decay. Near the boundary the derivatives of high-frequency modes can still be large, so interior smoothing should not be mistaken for a uniform derivative estimate independent of rough boundary data.

For smooth bounded domains and sufficiently regular data, Green's identity represents a Dirichlet Poisson solution as a volume source integral plus a boundary term involving the normal derivative of the Green function. With the positive minus-Laplacian convention, the Poisson weight is minus the outward source-normal derivative of G. Regularity is needed to interpret that derivative classically; rough domains often require harmonic measure or weak formulations instead. A boundary integral written without its measure and normal orientation is incomplete.

A Green function can fail to be unique or exist in the desired form if the operator and boundary condition have a kernel. Pure Neumann problems, for example, need a compatibility condition and normalization, so their Green functions require a corresponding constant correction. Symmetry of the zero-Dirichlet Green function follows from the self-adjoint setting, not from every variable-coefficient operator. The interval and disk formulas illustrate specific invertible Laplacian problems rather than a universal formula for all elliptic equations.