Geodesic
On the Solvability, in the Class of Polynomials, of the Dirichlet Problem for the Laplace Equation on an Arbitrary Polygon
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 102-114

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A constructive algorithm is developed that distinguishes between the cases when the solution to the Dirichlet problem for the Laplace equation is or is not a harmonic polynomial when the boundary values on the sides of an arbitrary polygon are specified by algebraic polynomials. 
@article{TM_2001_232_a10,
     author = {E. A. Volkov},
     title = {On the {Solvability,} in the {Class} of {Polynomials,} of the {Dirichlet} {Problem} for the {Laplace} {Equation} on an {Arbitrary} {Polygon}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {102--114},
     publisher = {mathdoc},
     volume = {232},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2001_232_a10/}
}
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E. A. Volkov. On the Solvability, in the Class of Polynomials, of the Dirichlet Problem for the Laplace Equation on an Arbitrary Polygon. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, harmonic analysis, and differential equations, Tome 232 (2001), pp. 102-114. http://geodesic.mathdoc.fr/item/TM_2001_232_a10/