THE ISOPERIMETRIC INEQUALITY AND ASSOCIATED BOUNDARY PROBLEMS

摘要

With the aid of this extremal property there is established also an integral inequality for simply connected rectifiable pieces of regular surfaces, Which in turn provides an elementary proof of the result of Carleman [Mathenatischs Zeitschrift 9 (1921),154-160] that the isoperimetric inequality persists for such pieces of minimal surfaces. The extremal property of the smallest proper value of a boundary problem related to (*) is show to afford a similar proof of the modified isoperimetric inequality for the simplest instance of a plane variational problem of "Dido type.”