By Isaac Chavel

The fundamental targets of the ebook are: (i) to introduce the topic to these drawn to gaining knowledge of it, (ii) to coherently current a few uncomplicated options and effects, at present utilized in the topic, to these operating in it, and (iii) to give a number of the effects which are beautiful of their personal correct, and which lend themselves to a presentation no longer overburdened with technical equipment.

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Then Jli E $3(M), and Jll,. ,$k may be chosen orthonormal in L2(M). As before, there exist al, . , U k , not all equal to zero, satisfying bl, . , k - 1 . j=1 Therefore the function k f C ajJlj = j=1 is orthogonal to b1,. , A- in L2(M),which implies k &l1f/lz I DCf,fl = vj$ vkllf j= 1 /I2 which is the claim. COROLLARY 1. If R E M , then for the Dirichlet eigenvalue problem on R, and any eigenvalue problem on M we have nk(n)2 (85) If M - a is open in M, then the inequality is strict. Domain monotonicity of eigenualues (vanishing Neumann data): Let be as above, and also assume a,, .

3) The functions 'p, determined in this manner are known to span L2(T) (Berger-Gauduchon-Mazet [l, pp. 146-1481). We remark that if given y,, . ,yk E r*,then the functions {by, : j = 1, . )k } are linearly independent. Indeed, if k = 1 then all is well. Should the I distinct elements y , , . ,yr E r*satisfy 1 for a given choice of complex numbers P,, . ,Pi, then, since 4Yr4Ys for all y,, y, E r*,we would have = 4Yyr+Y, 30 11. The Basic Examples Now calculate the Laplacian of both sides; then 1-1 0 = 1 BjIyj - ~lI'$yj-yr, j= 1 -- from which one easily concludes: fll = .

I, Chap. 51 and Coddington-Levinson [l, Chaps. 7, 81) yield that the collection of real A, for which there is a nontrivial solution of (32), with v = v,, satisfying the given boundary data, consists of a sequence 0s Al,1 < AI,2 < . - . T +a, and for each j = 1,2,. ,A,,j determines only a l-dimensional space of solutions. One easily checks, using integration-by-parts, that if j # k and q,j,T , k are solutions of (32), with v = vI, for A,,,, respectively, then q,j and T , k are orthogonal in the Lz space on (0,6) whose measure has the density S:-'(t) dt.