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This dissertation is devoted to the resolution of the Plateau problem in thecase of polygonal boundary curves in three-dimensional Euclidean space. Itrelies on the method developed by Ren\'e Garnier and published in 1928 in apaper which seems today to be totally forgotten. Garnier's approach is moregeometrical and constructive than the variational one, and it provides minimaldisks without branch point. However, it is sometimes really complicated, andeven obscure or incomplete. Following Garnier's initial ideas, we propose a newproof, which intends not only to be complete, but also simpler and modernerthan his one. This work mainly relies on a systematic use of Fuchsian systemsand on the relation that we establish between the reality of such systems andtheir monodromy. Garnier's method is based on the following fact: using thespinor Weierstrass representation for minimal surfaces, we can associate a realFuchsian second-order equation, defined on the Riemann sphere, with eachminimal disk with a polygonal boundary curve. The monodromy of the equation isdetermined by the oriented directions of the edges of the boundary. To solvethe Plateau problem, we are thus led to solve a Riemann-Hilbert problem. Wethen proceed in two steps: first, by means of isomonodromic deformations, weconstruct and describe the family of all minimal disks with a polygonalboundary curve of given oriented directions. Then we use this description tostudy the edges's lengths of their boundary curves, and we show that everypolygon is the boundary of a minimal disk.

Probl\`eme de Plateau, \'equations fuchsiennes et probl\`eme de Riemann-Hilbert