The application of isoperimetric inequalities for nonlinear eigenvalue problems

Our aim is to show the interplay between geometry analysis and applications of the theory of isoperimetric inequalities for some nonlinear problems. Reviewing the isoperimetric inequalities valid on Minkowskian plane we show that we can get estimations of physical quantities, namely, estimation on the first eigenvalue of nonlinear eigenvalue problems, on the basis of easily accessible geometrical data. Key-Words: nonlinear eigenvalue problems, isoperimetric inequalities, Minkowskian geometry 1 The classical isoperimetric inequality The classical isoperimetric inequality after which all such inequalities are named states that of all plane curves of given perimeter the circle encloses the largest area. This extremal property is expressed in the inequality: A L 4 2 π ≥ , (1) where A denotes the area of the domain and L the length of its boundary curve, and where equality holds only for circles. This inequality was known already to the Greeks. Pappus, in whose writings these results are preserved, attributes their discovery to Zenodorus. In their famous book Isoperimetric Inequalities in Mathematical Physics, Pólya and Szegő extended this notion to include inequalities for domain functionals, provided that the equality sign is attained for some domain or in the limit as the domain degenerates [15]. 2 Isoperimetric inequalities in “broader sense” There are several interesting and important geometrical and physical quantities depending on the shape and size of a curve: -the length of its perimeter, the area included, -the moment of inertia, with respect to the centroid, of a homogeneous plate bounded by the curve, -the torsional rigidity of an elastic beam the cross section of which is bounded by the given curve, -the principal frequency of a membrane of which the given curve is the rim, -the electrostatic capacity of a plate of the same shape and size, -and several other quantities. By the help of the isoperimetric inequalities we estimate physical quantities on the basis of easily accessible geometrical data. The study of isoperimetric inequalities in a broader sense began with the conjecture of St Venant in 1856, that of all cylindrical beams of given cross-sectional area the circular beam has the highest torsional rigidity. In 1877 Lord Rayleigh conjectured that of all vibrating elastic membranes of constant density and fixed area the circular membrane has the minimum principal frequency. He gave some evidence to support the conjecture. In 1903 H. Poincaré made the conjecture that of all solids of given volume the sphere has the minimum exterior electrostatic capacity. The proofs of these conjectures were given later. Around 1923 G. Faber [7] and E. Krahn [10] obtained independently the statement of Rayleigh. The proof were based on the introduction of a special system of curvilinear coordinates. G. Szegő and G. Pólya gave another proof by using the Steiner symmetrization [15]. In recent literature the statement of the Rayleigh conjecture is usually referred to as the Faber-Krahn inequality. This property is expressed by the inequality