Harmonic functions and Dirichlet problems

A real-valued twice continuously differentiable function u is harmonic on an open set when its Laplacian vanishes. If u is the real part of a holomorphic f=u+iv, the Cauchy–Riemann equations and commuting mixed derivatives give that equation. Holomorphic functions are smooth by Cauchy theory, so the derivative interchange is justified. The imaginary part is harmonic as well. Harmonicity therefore captures part of the structure of a holomorphic function without recording the full complex-valued map.

Conversely, the equations for a conjugate are $v_x=-u_y$ and $v_y=u_x$. Harmonicity makes this differential form locally integrable, so a conjugate exists on small disks. On a simply connected domain the path integrals can define a global conjugate, unique up to an additive real constant. On the punctured plane, log|z| is harmonic but a global conjugate would provide a single-valued argument, which is impossible. A local derivative condition does not remove a global period obstruction.

Give a continuous real function F on the unit circle. Its Poisson integral defines a harmonic function for r<1, extends continuously to the closed disk, and equals F on the boundary. The kernel is positive and has integral 2π, so the construction averages boundary values with location-dependent weights. At the center those weights are uniform, recovering the mean-value property. Toward a boundary point they concentrate near that point, which explains why continuity yields the stated boundary limit.

Uniqueness follows from the maximum principle: the difference of two harmonic continuous-closure solutions has zero boundary values and therefore has neither a positive maximum nor a negative minimum inside. Boundedness of the domain and continuity up to the boundary matter for this argument. On other domains, existence can depend on boundary regularity and the meaning of boundary data. Do not import the disk's clean conclusion into a region with unexamined boundary behavior.

A boundary cosine of frequency n extends as rⁿcos(nθ), and a boundary sine extends as rⁿsin(nθ). These are real or imaginary parts of zⁿ and hence harmonic. Finite Fourier data can therefore be extended term by term without a convergence complication. Higher angular frequencies are attenuated more strongly at a fixed interior radius, expressing a spatial smoothing effect. The constant mode remains constant and determines the center value.

This radial attenuation should not be confused with heat evolution, whose Fourier modes decay as the exponential of minus diffusivity times n² times time. Both favor low frequencies, but they solve different equations with different independent variables and factors. Checking the governing equation prevents one appealing smoothing animation from being mistaken for another solver. For a proposed disk solution, verify harmonicity, boundary matching, and continuity separately; uniqueness then certifies that no second solution with the same conditions has been overlooked.