Geodesic
Mappings by the Solutions of Second-Order Elliptic Equations
Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 718-733

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The properties of mappings by the solutions of second-order elliptic partial differential equations in the plane are studied. We obtain conditions on a function, continuous on the unit circle, that are sufficient for the solution of the Dirichlet problem in the open unit disk for the given equation with the given boundary function to be a homeomorphism between the open unit disk and a Jordan simply connected domain. The properties of the zeros of the solutions of the given equations are also studied. In particular, an analog of the main theorem of algebra is proved for polynomial solutions.
Keywords: elliptic partial differential equation, Dirichlet problem, Jordan simply connected domain, Fourier series. 
@article{MZM_2014_95_5_a7,
     author = {A. B. Zaitsev},
     title = {Mappings by the {Solutions} of {Second-Order} {Elliptic} {Equations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {718--733},
     publisher = {mathdoc},
     volume = {95},
     number = {5},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a7/}
}
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A. B. Zaitsev. Mappings by the Solutions of Second-Order Elliptic Equations. Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 718-733. http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a7/