# Examples of 2D Harmonic Functions

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A function is harmonic on a domain if it satisfies the Laplace equation in the interior of . A remarkable property of harmonic functions is that they are uniquely defined by their values on the boundary of . Geometrically, this means that, given any smooth 3D curve defined on the boundary of , there exists a unique harmonic surface (i.e. a surface where is harmonic) whose boundary is the given curve.

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Contributed by: Baihe Duan (January 2019)

Based on: an undergraduate research project at the Illinois Geometry Lab by Yuheng Chang, Baihe Duan, Yirui Luo, Yitao Meng, Cameron Nachreiner and Yiyin Shen, directed by A. J. Hildebrand

Open content licensed under CC BY-NC-SA

## Details

Specifying the Boundary Curve

Using polar coordinates, a closed curve defined on the boundary of the unit disk can be constructed using a function that is periodic in . Therefore, specifying a boundary curve is equivalent to specifying such a periodic function . For the custom boundary function, a periodic Hermite interpolation of order 2 is used to match the given points (together with the points and ) to a piecewise polynomial function with period .

Computing the Harmonic Surface

The values of in the interior of are computed by numerically solving the Laplace equation with the specified boundary condition.

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