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A Unified Approach to Estimation of Lower Bounds for the First Eigenvalue of Several Elliptic Boundary Value Problems

✍ Scribed by Klaus Gürlebeck; Wolfgang Sprössig

Publisher
John Wiley and Sons
Year
1987
Tongue
English
Weight
729 KB
Volume
131
Category
Article
ISSN
0025-584X

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✦ Synopsis

History of the problem

While it is relatively easy to get estimations of upper bounds for eigerivalues of elliptic differential operators with the well-known methods of J. W. S. RAYLEIGH and W. RITZ, the estimation of lower bounds is much more complicated. Estimations of lower bounds for eigenvnlues were first established in 1936 by E. TREFFTZ/F. A. WILLERS [31], stimulating by problems in engineering. I n 1948 N. J. LEHWN [ 171 proposed a method for the determination of lower bounds for the eigenvalues of special elliptic problems in the plane. The development of this method has been carried on by L. COLLATZ [6], J. ALBRECHT [ 11, F. GOERISCH/H. HAUNHORST [9], and others. Since 1960 essential results have been obtained using several geometrical considerations. For the case of convex domains Q c RZ J. HERSCH El21 in 1960 proved the following estimation of the first eigenvalue of the DrrtIcmET-problem for the LAPLACE-equation I n this formula e means the radius of the largest disc contained in the domain G. R. OSSERMA" [21] on the basis of an inequality stated by J. CEEEOER [5] in 1970 showed the estimation for k-fold connected domains. This result was essentially improved by M. TAYLOR [30] in 1979. I n the case k 2 2 he showed the existence of a constant c with

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