Explicit 0 ( it 2 ) Bounds on the Eigenvalues of the Half - L * By Blair

0. Summary and Survey. This paper is concerned with obtaining strict upper and lower bounds on the eigenvalues of a particular nontrivial convex membrane, the half-L, which is fixed or free at the boundary. The upper bound is obtained from a matrix eigenvalue calculation; the matrix problem may be regarded as a difference scheme although it is derived using piecewise linear functions in a Rayleigh quotient. The lower bound is than calculated from the upper bound using an elementary formula. The validity of this formula is proved by extensions of Weinberger's techniques [3]. Difficulties encountered in determining similar results for the nonconvex L-shaped membrane are indicated. A numerical example illustrates the results. An appendix contains some pointwise bounds on normalized eigenfunctions. An annotated survey of the literature concerned with estimating the eigenvalues of the Laplacian is contained in [4]; many other references are found in [10]. The use of piecewise linear functions in variational principles is discussed in [1], [2], [5]; piecewise bilinear in [6], [7], [9, pp. 331-334]. A general discussion of Rayleigh-Ritz eigenvalues is found in [8]. Much of the literature concerned with strict bounds on the eigenvalues seems to use the eigenvalues of the discrete Laplacian or a related matrix rather than the eigenvalues associated with the Rayleigh-Ritz method. The lower bounds of [19], [3], [13], and the simultaneous two-sided bounds in [10] are 0(a) bounds as a result of embedding a general region in a union of squares [9, p. 339], [4, p. 30]. It is not clear how the lower bound in [14] behaves as h —» 0. (See also [4, pp. 30-31].) Other results have been asymptotic: the discrete eigenvalue is the continuous one (in certain cases) except for a term yh2 + oQi2), y unknown [4], [9]; a result quite useful in extrapolation to zero.