## On the fundamental eigenvalue of a solid |

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associated with j.(B Bessel Function boundary problem bounded region chapter characteristic equation closed set computed constant continuous first derivatives continuous function Courant Definition II.1 Definition V.6 differential equation Dinghas eigenfunction associated eigenvector En+1 endpoints equilateral triangle Euclidean Euclidean plane Euler equation Example II.1 Figure f dV f f dVn f is piecewise f satisfies f vanishes finite number function defined Given a solid hypercube III.1.6 become intersects Isoperimetric Inequalities Lemma VII.2 Let f Math measurable function mesh MINIMAL PROPERTY orthogonal piecewise smooth plane region Polya and G polyhedral function polyhedron Proof properties of Steiner Q.E.D. Section Q.E.D. Theorem radius rectangle with sides respect semicircle simplexes smallest fundamental eigenvalue sphere Steiner symmetrization sufficiently small surface area Szego Theorem III.3 Theorem V•5 values vertex Vf)2 dVn Vn(B Vn(K Xa cos X(l Xa sin X(l xn+1 yields zero