A beautiful inequality by Saint-Venant and Pólya revisited. (English) Zbl 1510.35013
Summary: In mathematical physics and beyond, one encounters many beautiful inequalities that relate geometric or physical quantities describing the shape or size of a set. Such isoperimetric inequalities often have a long history and many important applications. For instance, the eponymous and most classical of all isoperimetric inequalities was known already in antiquity. It asserts that among all closed planar curves of a given length, the circles with perimeter equal to that length, and only they, enclose the largest area. Though not nearly as well-known, an isoperimetric inequality conjectured by Saint-Venant in the 1850s and first proved by Pólya almost a century later, is also very beautiful and important. By presenting a short proof as well as two simple physical interpretations, this article illustrates why the result deserves to be cherished by every student of applied analysis.
MSC:
35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
26D10 | Inequalities involving derivatives and differential and integral operators |
97M50 | Physics, astronomy, technology, engineering (aspects of mathematics education) |
35J25 | Boundary value problems for second-order elliptic equations |
74B05 | Classical linear elasticity |
76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
Keywords:
isoperimetric inequalityReferences:
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