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| The ``\PMlinkescapetext{Plateau}'s Problem'' is the problem of finding the surface with minimal area among all surfaces wich have the same prescribed boundary. |
The ``\PMlinkescapetext{Plateau}'s Problem'' is the problem of finding the surface with minimal area among all surfaces wich have the same prescribed boundary. |
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| This problem is named after the Belgian physicist Joseph \PMlinkescapetext{Plateau} (1801-1883) who experimented with soap films. As a matter of fact if you take a wire (which represents a closed curve in three-dimensional space) and dip it in a solution of soapy water, you obtain a soapy surface which has the wire as boundary. |
This problem is named after the Belgian physicist Joseph \PMlinkescapetext{Plateau} (1801-1883) who experimented with soap films. As a matter of fact if you take a wire (which represents a closed curve in three-dimensional space) and dip it in a solution of soapy water, you obtain a soapy surface which has the wire as boundary. |
| It turns out that this surface has the minimal area among all surfaces with the same boundary, so the soap film is a solution to the \PMlinkescapetext{Plateau}'s Problem. |
It turns out that this surface has the minimal area among all surfaces with the same boundary, so the soap film is a solution to the \PMlinkescapetext{Plateau}'s Problem. |
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| Jesse Douglas (1897-1965) solved the problem by proving the existence of such minimal surfaces. The solution to the problem is achieved by finding an harmonic and conformal parameterization of the surface. |
Jesse Douglas (1897-1965) solved the problem by proving the existence of such minimal surfaces. The solution to the problem is achieved by finding an harmonic and conformal parameterization of the surface. |
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| The extension of the problem to higher dimensions (i.e. for $k$-dimensional surfaces in $n$-dimensional space) turns out to be much more difficult to study. |
The extension of the problem to higher dimensions (i.e. for $k$-dimensional surfaces in $n$-dimensional space) turns out to be much more difficult to study. |
| Moreover while the solutions to the original problem are always regular it turns out that the solutions to the extended problem may have singularities if |
Moreover while the solutions to the original problem are always regular it turns out that the solutions to the extended problem may have singularities if |
| $n\ge 8$. |
$n\ge 8$. |
| To solve the extended problem the theory of perimeters (De Giorgi) for |
To solve the extended problem the theory of perimeters (De Giorgi) for |
| boundaries and the theory of rectifiable |
boundaries and the theory of rectifiable |
| currents (Federer and Fleming) has been developed. |
currents (Federer and Fleming) has been developed. |
