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Menahem Max Schiffer: Selected Papers Volume 2 by Peter Duren, Lawrence Zalcman

By Peter Duren, Lawrence Zalcman

M. M. Schiffer, the dominant determine in geometric functionality thought within the moment 1/2 the 20th century, used to be a mathematician of remarkable breadth, whose paintings ranged over such components as univalent features, conformal mapping, Riemann surfaces, partial differential equations, power idea, fluid dynamics, and the idea of relativity. he's most sensible remembered for the robust variational equipment he constructed and utilized to extremal difficulties in a wide selection of clinical fields.

Spanning seven many years, the papers accumulated in those volumes signify a few of Schiffer's so much enduring strategies. professional commentaries supply beneficial heritage and survey next advancements. additionally incorporated are a whole bibliography and a number of other appreciations of Schiffer's impact through collaborators and different admirers.

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1 rr ;JJJsi1. 13) in this case and that it is of infinite degeneracy. Our variational theory does not work in this exceptional case. However, let \z0 \ 1 and C* be the image of \z\=1 under the variation (1). We define its eigen function w;(z) and by (3) a function m~(z) which is regular analytic in D. It satisfies the integral equation (5) which, in view of (32), can be brought into the simple form > a r;=zo+---. 'REDHOLM EIGEN VALDES OF PLANE DOMAINS (34) m~(z)=~ M~(z) 1211 ; if we choose the right constant of integration in the definition of M,(z), we can integrate (33) to the identity (35) where (36) is a linear function of z.

A generalization of most concepts to the physically more interesting case of three dimensions is easily done. 3. The variation of the eigen values. The variation of the eigen values Av under a variation of the curve C can be determined by using the variational theory of the Green's function and of the various kernel functions connected with it [3]. In this paper we wish to give a straightforward and elementary derivation of the variational formulas. Let z0 be an arbitrary fixed point in D and consider the mapping ('tl ) z*=z+ __ ("¥_ z-zo 19 • THE FREDHOLM EIGEN VALUES OF PLANE DOMAINS 1205 For small enough a this will be a univalent mapping of C into a new smooth curve C*.

V(()ds; rr = {~,(z) for z E D hv(z) a for z E D. The set of harmonic functions hv(z) and hv(z) can be interpreted as the potential due to a double layer of logarithmic charges, spread along C with the density (A,/rr)rf\((). Hence, the well known discontinuity character of such potentials leads to the boundary relations at each point and ( 4') where n denotes the normal of C pointing into 15. The Fredholm eigen value problem may thus be formulated as the following question of potential theory which is of interest by itself : 45 FREDHOLM EIGENVALUES OF MULTIPLY-CONNECTED DOMAINS 215 To determine a harmonic function h in i5 and a set of harmonic functions h in D which have equal normal derivatives and proportional boundary values on C!.

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